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Big Ideas Math Book Algebra 2 Answer Key Chapter 3 Quadratic Equations and Complex Numbers

Download the Topic-wise BigIdeas math book textbook solutions of algebra 2 ch 3 Quadratic Equations and Complex Numbers pdf for free of charge and practice well at any time and anywhere you wish. Once you start preparing the concepts of chapter 3 from BIM Textbook Solution Key, you can stand out from the crowd and become a topper in the class. Just tap on the available links below and Download Big Ideas Math Book Algebra 2 Ch 3 Answers Key freely. These solutions of BIM Algebra 2 Quadratic Equations and Complex Numbers are prepared by the subject experts according to the guidelines of the Common core standards. 

• Quadratic Equations and Complex Numbers Maintaining Mathematical Proficiency – Page 91
• Quadratic Equations and Complex Numbers Mathematical Practices – Page 92
• Lesson 3.1 Solving Quadratic Equations – Page(94-102)
• Solving Quadratic Equations 3.1 Exercises – Page(99-102)
• Lesson 3.2 Complex Numbers – Page(104-110)
• Complex Numbers 3.2 Exercises – Page(108-110)
• Lesson 3.3 Completing the Square – Page(112-118)
• Completing the Square 3.3 Exercises – Page(116-118)
• Quadratic Equations and Complex Numbers Study Skills: Creating a Positive Study Environment – Page 119
• Quadratic Equations and Complex Numbers 3.1–3.3 Quiz – Page 120
• Lesson 3.4 Using the Quadratic Formula – Page(122-130)
• Using the Quadratic Formula 3.4 Exercises – Page(127-130)
• Lesson 3.5 Solving Nonlinear Systems – Page(132-138)
• Solving Nonlinear Systems 3.5 Exercises – Page(136-138)
• Lesson 3.6 Quadratic Inequalities – Page(140-146)
• Quadratic Inequalities 3.6 Exercises – Page(144-146)
• Quadratic Equations and Complex Numbers Performance Task – Page 147
• Quadratic Equations and Complex Numbers Chapter Review – Page(148-150)
• Quadratic Equations and Complex Numbers Chapter Test – Page 151
• Quadratic Equations and Complex Numbers Cumulative Assessment – Page(152-153)

Quadratic Equations and Complex Numbers Maintaining Mathematical Proficiency

Simplify the expression.
Question 1.
\(\sqrt{27}\)
Answer:

Question 2.
–\(\sqrt{112}\)
Answer:

Question 3.
\(\sqrt{\frac{11}{64}}\)
Answer:

Question 4.
\(\sqrt{\frac{147}{100}}\)
Answer:

Question 5.
\(\sqrt{\frac{18}{49}}\)
Answer:

Question 6.
–\(\sqrt{\frac{65}{121}}\)
Answer:

Question 7.
–\(\sqrt{80}\)
Answer:

Question 8.
\(\sqrt{32}\)
Answer:

Factor the polynomial.
Question 9.
x² − 36
Answer:

Question 10.
x² − 9
Answer:

Question 11.
4x² − 25
Answer:

Question 12.
x² − 22x + 121
Answer:

Question 13.
x² + 28x + 196
Answer:

Question 14.
49x² + 210x + 225
Answer:

Question 15.
ABSTRACT REASONING
Determine the possible integer values of a and c for which the trinomial ax²+ 8x+c is factorable using the Perfect Square Trinomial Pattern. Explain your reasoning.
Answer:

Quadratic Equations and Complex Numbers Mathematical Practices

Mathematically proficient students recognize the limitations of technology

Monitoring Progress

Question 1.
Explain why the second viewing window in Example 1 shows gaps between the upper and lower semicircles, but the third viewing window does not show gaps.
Answer:

Use a graphing calculator to draw an accurate graph of the equation. Explain your choice of viewing window.
Question 2.
y = \(\sqrt{x^{2}-1.5}\)
Answer:

Question 3.
y = \(\sqrt{x-2.5}\)
Answer:

Question 4.
x² + y²= 12.25
Answer:

Question 5.
x² + y² = 20.25
Answer:

Question 6.
x² + 4y² = 12.25
Answer:

Question 7.
4x² + y² = 20.25
Answer:

Lesson 3.1 Solving Quadratic Equations

Essential Question How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

EXPLORATION 1

Matching a Quadratic Function with Its Graph
Work with a partner. Match each quadratic function with its graph. Explain your reasoning. Determine the number of x-intercepts of the graph.
a. f(x) = x² − 2x
b. f(x) = x² − 2x + 1
c. f(x) = x² − 2x + 2
d. f(x) = −x² + 2x
e. f(x) = −x² + 2x − 1
f. f(x) = −x² + 2x − 2

EXPLORATION 2

Solving Quadratic Equations
Work with a partner. Use the results of Exploration 1 to find the real solutions (if any) of each quadratic equation.
a. x² − 2x = 0
b. x² − 2x + 1 = 0
c. x² − 2x + 2 = 0
d. −x² + 2x = 0
e. −x² + 2x − 1 = 0
f. −x² + 2x − 2 = 0

Communicate Your Answer

Question 3.
How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

Answer:

Question 4.
How many real solutions does the quadratic equation x² + 3x + 2 = 0 have? How do you know? What are the solutions?
Answer:

Monitoring Progress

Solve the equation by graphing.
Question 1.
x² − 8x + 12 = 0
Answer:

Question 2.
4x² − 12x + 9 = 0
Answer:

Question 3.
\(\frac{1}{2}\)x² = 6x − 20
Answer:

Solve the equation using square roots.
Question 4.
\(\frac{2}{3}\)x² + 14 = 20
Answer:

Question 5.
−2x² + 1 = −6
Answer:

Question 6.
2(x − 4)² = −5
Answer:

Solve the equation by factoring.
Question 7.
x² + 12x + 35 = 0
Answer:

Question 8.
3x² − 5x = 2
Answer:

Find the zero(s) of the function.
Question 9.
f(x) = x² − 8x
Answer:

Question 10.
f(x) = 4x² + 28x + 49
Answer:

Question 11.
WHAT IF?
The magazine initially charges $21 per annual subscription. How much should the magazine charge to maximize annual revenue? What is the maximum annual revenue?
Answer:

Question 12.
WHAT IF?
The egg container is dropped from a height of 80 feet. How does this change your answers in parts (a) and (b)?
Answer:

Solving Quadratic Equations 3.1 Exercises

Vocabulary and Core Concept Check
Question 1.
WRITING
Explain how to use graphing to find the roots of the equation ax² + bx + c = 0.
Answer:

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, solve the equation by graphing.
Question 3.
x² + 3x + 2 = 0
Answer:

Question 4.
−x² + 2x + 3 = 0
Answer:

Question 5.
y = x² − 9
Answer:

Question 6.
−8 = −x² − 4
Answer:

Question 7.
8x = −4 − 4x²
Answer:

Question 8.
3x² = 6x − 3
Answer:

Question 9.
7 = −x² − 4x
Answer:

Question 10.
2x = x² + 2
Answer:

Question 11.
\(\frac{1}{5}\)x² + 6 = 2x
Answer:

Question 12.
3x = \(\frac{1}{4}\)x² + 5
Answer:

In Exercises 13–20, solve the equation using square roots.
Question 13.
s² = 144
Answer:

Question 14.
a² = 81
Answer:

Question 15.
(z − 6)² = 25
Answer:

Question 16.
(p − 4)² = 49
Answer:

Question 17.
4(x − 1)² + 2 = 10
Answer:

Question 18.
2(x + 2)² − 5 = 8
Answer:

Question 19.
\(\frac{1}{2}\)r² − 10 = \(\frac{3}{2}\)r²
Answer:

Question 20.
\(\frac{1}{5}\)x² + 2 = \(\frac{3}{5}\)x²
Answer:

Question 21.
ANALYZING RELATIONSHIPS
Which equations have roots that are equivalent to the x-intercepts of the graph shown?

A. −x² − 6x − 8 = 0
B. 0 = (x + 2)(x + 4)
C. 0 = −(x + 2)² + 4
D. 2x² − 4x − 6 = 0
E. 4(x + 3)² − 4 = 0
Answer:

Question 22.
ANALYZING RELATIONSHIPS
Which graph has x-intercepts that are equivalent to the roots of the equation (x − \(\frac{3}{2}\))² = \(\frac{25}{4}\)? Explain your reasoning.

Answer:

ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in solving the equation.
Question 23.

Answer:

Question 24.

Answer:

Question 25.
OPEN-ENDED
Write an equation of the form x² = d that has (a) two real solutions, (b) one real solution, and (c) no real solution.
Answer:

Question 26.
ANALYZING EQUATIONS
Which equation has one real solution? Explain.
A. 3x² + 4 = −2(x² + 8)
B. 5x² − 4 = x² − 4
C. 2(x + 3)² = 18
D. \(\frac{3}{2}\)x² − 5 = 19
Answer:

In Exercises 27–34, solve the equation by factoring.
Question 27.
0 = x² + 6x + 9
Answer:

Question 28.
0 = z² − 10z + 25
Answer:

Question 29.
x² − 8x = −12
Answer:

Question 30.
x² − 11x = −30
Answer:

Question 31.
n² − 6n = 0
Answer:

Question 32.
a² − 49 = 0
Answer:

Question 33.
2w² − 16w = 12w − 48
Answer:

Question 34.
−y + 28 + y² = 2y + 2y²
Answer:

MATHEMATICAL CONNECTIONS In Exercises 35–38, find the value of x.
Question 35.
Area of rectangle = 36

Answer:

Question 36.
Area of circle = 25π

Answer:

Question 37.
Area of triangle = 42

Answer:

Question 38.
Area of trapezoid = 32

Answer:

In Exercises 39–46, solve the equation using any method. Explain your reasoning.
Question 39.
u² = −9u
Answer:

Question 40.
\(\frac{t^{2}}{20}\) + 8 = 15
Answer:

Question 41.
−(x + 9)² = 64
Answer:

Question 42.
−2(x + 2)² = 5
Answer:

Question 43.
7(x − 4)² − 18 = 10
Answer:

Question 44.
t² + 8t + 16 = 0
Answer:

Question 45.
x² + 3x + \(\frac{5}{4}\) = 0
Answer:

Question 46.
x² − 1.75 = 0.5
Answer:

In Exercises 47–54, find the zero(s) of the function.
Question 47.
g(x) = x² + 6x + 8
Answer:

Question 48.
f(x) = x² − 8x + 16
Answer:

Question 49.
h(x) = x² + 7x − 30
Answer:

Question 50.
g(x) = x² + 11x
Answer:

Question 51.
f(x) = 2x² − 2x − 12
Answer:

Question 52.
f(x) = 4x² − 12x + 9
Answer:

Question 53.
g(x) = x² + 22x + 121
Answer:

Question 54.
h(x) = x² + 19x + 84
Answer:

Question 55.
REASONING
Write a quadratic function in the form f(x) = x² + bx + c that has zeros 8 and 11.
Answer:

Question 56.
NUMBER SENSE
Write a quadratic equation in standard form that has roots equidistant from 10 on the number line.
Answer:

Question 57.
PROBLEM SOLVING
A restaurant sells 330 sandwiches each day. For each $0.25 decrease in price, the restaurant sells about 15 more sandwiches. How much should the restaurant charge to maximize daily revenue? What is the maximum daily revenue?

Answer:

Question 58.
PROBLEM SOLVING
An athletic store sells about 200 pairs of basketball shoes per month when it charges $120 per pair. For each $2 increase in price, the store sells two fewer pairs of shoes. How much should the store charge to maximize monthly revenue? What is the maximum monthly revenue?
Answer:

Question 59.
MODELING WITH MATHEMATICS
Niagara Falls is made up of three waterfalls. The height of the Canadian Horseshoe Falls is about 188 feet above the lower Niagara River. A log falls from the top of Horseshoe Falls.
a. Write a function that gives the height h (in feet) of the log after t seconds. How long does the log take to reach the river?
b. Find and interpret h(2) − h(3).
Answer:

Question 60.
MODELING WITH MATHEMATICS
According to legend, in 1589, the Italian scientist Galileo Galilei dropped rocks of different weights from the top of the Leaning Tower of Pisa to prove his conjecture that the rocks would hit the ground at the same time. The height h (in feet) of a rock after t seconds can be modeled by h(t) = 196 − 16t².

a. Find and interpret the zeros of the function. Then use the zeros to sketch the graph.
b. What do the domain and range of the function represent in this situation?
Answer:

Question 61.
PROBLEM SOLVING
You make a rectangular quilt that is 5 feet by 4 feet. You use the remaining 10 square feet of fabric to add a border of uniform width to the quilt. What is the width of the border?

Answer:

Question 62.
MODELING WITH MATHEMATICS
You drop a seashell into the ocean from a height of 40 feet. Write an equation that models the height h (in feet) of the seashell above the water after t seconds. How long is the seashell in the air?
Answer:

Question 63.
WRITING
The equation h = 0.019s² models the height h (in feet) of the largest ocean waves when the wind speed is s knots. Compare the wind speeds required to generate 5-foot waves and 20-foot waves.

Answer:

Question 64.
CRITICAL THINKING
Write and solve an equation to find two consecutive odd integers whose product is 143.
Answer:

Question 65.
MATHEMATICAL CONNECTIONS
A quadrilateral is divided into two right triangles as shown in the figure. What is the length of each side of the quadrilateral?

Answer:

Question 66.
ABSTRACT REASONING
Suppose the equation ax² + bx + c = 0 has no real solution and a graph of the related function has a vertex that lies in the second quadrant.
a. Is the value of a positive or negative? Explain your reasoning.
b. Suppose the graph is translated so the vertex is in the fourth quadrant. Does the graph have any x-intercepts? Explain.
Answer:

Question 67.
REASONING
When an object is dropped on any planet, its height h (in feet) after t seconds can be modeled by the function h = −\(\frac{g}{2}\)t² + h₀, where h₀ is the object’s initial height and g is the planet’s acceleration due to gravity. Suppose a rock is dropped from the same initial height on the three planets shown. Make a conjecture about which rock will hit the ground first. Justify your answer.

Answer:

Question 68.
PROBLEM SOLVING
A café has an outdoor, rectangular patio. The owner wants to add 329 square feet to the area of the patio by expanding the existing patio as shown. Write and solve an equation to find the value of x. By what distance should the patio be extended?

Answer:

Question 69.
PROBLEM SOLVING
A flea can jump very long distances. The path of the jump of a flea can be modeled by the graph of the function y = −0.189x² + 2.462x, where x is the horizontal distance (in inches) and y is the vertical distance (in inches). Graph the function. Identify the vertex and zeros and interpret their meanings in this situation.
Answer:

Question 70.
HOW DO YOU SEE IT?
An artist is painting a mural and drops a paintbrush. The graph represents the height h (in feet) of the paintbrush after t seconds.

a. What is the initial height of the paintbrush?
b. How long does it take the paintbrush to reach the ground? Explain.
Answer:

Question 71.
MAKING AN ARGUMENT
Your friend claims the equation x² + 7x =−49 can be solved by factoring and has a solution of x = 7. You solve the equation by graphing the related function and claim there is no solution. Who is correct? Explain.
Answer:

Question 72.
ABSTRACT REASONING
Factor the expressions x² − 4 and x² − 9. Recall that an expression in this form is called a difference of two squares. Use your answers to factor the expression x² − a². Graph the related function y = x² − a². Label the vertex, x-intercepts, and axis of symmetry.
Answer:

Question 73.
DRAWING CONCLUSIONS
Consider the expression x² + a², where a > 0.
a. You want to rewrite the expression as (x + m)(x + n). Write two equations that m and n must satisfy.
b. Use the equations you wrote in part (a) to solve for m and n. What can you conclude?
Answer:

Question 74.
THOUGHT PROVOKING
You are redesigning a rectangular raft. The raft is 6 feet long and 4 feet wide. You want to double the area of the raft by adding to the existing design. Draw a diagram of the new raft. Write and solve an equation you can use to find the dimensions of the new raft.

Answer:

Question 75.
MODELING WITH MATHEMATICS
A high school wants to double the size of its parking lot by expanding the existing lot as shown. By what distance x should the lot be expanded?

Answer:

Maintaining Mathematical Proficiency

Find the sum or difference.
Question 76.
(x² + 2) + (2x² − x)
Answer:

Question 77.
(x³ + x² − 4) + (3x² + 10)
Answer:

Question 78.
(−2x + 1) − (−3x² + x)
Answer:

Question 79.
(−3x³ + x² − 12x) − (−6x² + 3x − 9)
Answer:

Find the product.
Question 80.
(x + 2)(x − 2)
Answer:

Question 81.
2x(3 − x + 5x²)
Answer:

Question 82.
(7 − x)(x − 1)
Answer:

Question 83.
11x(−4x² + 3x + 8)
Answer:

Lesson 3.2 Complex Numbers

Essential Question What are the subsets of the set of complex numbers?
In your study of mathematics, you have probably worked with only real numbers, which can be represented graphically on the real number line. In this lesson, the system of numbers is expanded to include imaginary numbers. The real numbers and imaginary numbers compose the set of complex numbers.

EXPLORATION 1

Classifying Numbers
Work with a partner. Determine which subsets of the set of complex numbers contain each number.
a. \(\sqrt{9}\)
b. \(\sqrt{0}\)
c. −\(\sqrt{4}\)
d. \(\sqrt{\frac{4}{9}}\)
e. \(\sqrt{2}\)
f. \(\sqrt{-1}\)

EXPLORATION 2

Complex Solutions of Quadratic Equations

Work with a partner. Use the definition of the imaginary unit i to match each quadratic equation with its complex solution. Justify your answers.
a. x² − 4 = 0
b. x² + 1 = 0
c. x² − 1 = 0
d. x² + 4 = 0
e. x² − 9 = 0
f. x² + 9 = 0
A. i
B. 3i
C. 3
D. 2i
E. 1
F. 2

Communicate Your Answer

Question 3.
What are the subsets of the set of complex numbers? Give an example of a number in each subset.
Answer:

Question 4.
Is it possible for a number to be both whole and natural? natural and rational? rational and irrational? real and imaginary? Explain your reasoning.
Answer:

Monitoring Progress

Find the square root of the number.
Question 1.
\(\sqrt{-4}\)
Answer:

Question 2.
\(\sqrt{-12}\)
Answer:

Question 3.
−\(\sqrt{-36}\)
Answer:

Question 4.
2\(\sqrt{-54}\)
Answer:

Find the values of x and y that satisfy the equation.
Question 5.
x + 3i = 9 − yi
Answer:

Question 6.
9 + 4yi = −2x + 3i
Answer:

Question 7.
WHAT IF?
In Example 4, what is the impedance of the circuit when the capacitor is replaced with one having a reactance of 7 ohms?
Answer:

Perform the operation. Write the answer in standard form.
Question 8.
(9 − i ) + (−6 + 7i )
Answer:

Question 9.
(3 + 7i ) − (8 − 2i )
Answer:

Question 10.
−4 − (1 + i) − (5 + 9i)
Answer:

Question 11.
(−3i)(10i)
Answer:

Question 12.
i(8 − i)
Answer:

Question 13.
(3 + i)(5 −i)
Answer:

Solve the equation.
Question 14.
x² = −13
Answer:

Question 15.
x²= −38
Answer:

Question 16.
x² + 11 = 3
Answer:

Question 17.
x² − 8 = −36
Answer:

Question 18.
3x² − 7 = −31
Answer:

Question 19.
5x² + 33 = 3
Answer:

Find the zeros of the function.
Question 20.
f(x) = x² + 7
Answer:

Question 21.
f(x) = −x² − 4
Answer:

Question 22.
f(x) = 9x² + 1
Answer:

Complex Numbers 3.2 Exercises

Vocabulary and Core Concept Check
Question 1.
VOCABULARY
What is the imaginary unit i defined as and how can you use i?
Answer:

Question 2.
COMPLETE THE SENTENCE
For the complex number 5 + 2i, the imaginary part is ____ and the real part is ____.
Answer:

Question 3.
WRITING
Describe how to add complex numbers.
Answer:

Question 4.
WHICH ONE DOESN’T BELONG?
Which number does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–12, find the square root of the number.
Question 5.
\(\sqrt{-36}\)
Answer:

Question 6.
\(\sqrt{-64}\)
Answer:

Question 7.
\(\sqrt{-18}\)
Answer:

Question 8.
\(\sqrt{-24}\)
Answer:

Question 9.
2\(\sqrt{-16}\)
Answer:

Question 10.
−3\(\sqrt{-49}\)
Answer:

Question 11.
−4\(\sqrt{-32}\)
Answer:

Question 12.
6\(\sqrt{-63}\)
Answer:

In Exercises 13–20, find the values of x and y that satisfy the equation.
Question 13.
4x + 2i = 8 + yi
Answer:

Question 14.
3x + 6i = 27 + yi
Answer:

Question 15.
−10x + 12i = 20 + 3yi
Answer:

Question 16.
9x − 18i = −36 + 6yi
Answer:

Question 17.
2x − yi = 14 + 12i
Answer:

Question 18.
−12x + yi = 60 − 13i
Answer:

Question 19.
54 − \(\frac{1}{7}\)yi = 9x− 4i
Answer:

Question 20.
15 − 3yi = \(\frac{1}{2}\)x + 2i
Answer:

In Exercises 21–30, add or subtract. Write the answer in standard form.
Question 21.
(6 − i) + (7 + 3i)
Answer:

Question 22.
(9 + 5i) + (11 + 2i )
Answer:

Question 23.
(12 + 4i) − (3 − 7i)
Answer:

Question 24.
(2 − 15i) − (4 + 5i)
Answer:

Question 25.
(12 − 3i) + (7 + 3i)
Answer:

Question 26.
(16 − 9i) − (2 − 9i)
Answer:

Question 27.
7 − (3 + 4i) + 6i
Answer:

Question 28.
16 − (2 − 3i) − i
Answer:

Question 29.
−10 + (6 − 5i) − 9i
Answer:

Question 30.
−3 + (8 + 2i) + 7i
Answer:

Question 31.
USING STRUCTURE
Write each expression as a complex number in standard form.
a. \(\sqrt{-9}+\sqrt{-4}-\sqrt{16}\)
b. \(\sqrt{-16}+\sqrt{8}+\sqrt{-36}\)
Answer:

Question 32.
REASONING
The additive inverse of a complex number z is a complex number z_a such that z + z_a = 0. Find the additive inverse of each complex number.
a. z = 1 + i
b. z = 3 − i
c. z = −2 + 8i
Answer:

In Exercises 33–36, find the impedance of the series circuit.
Question 33.

Answer:

Question 35.

Answer:

In Exercises 37–44, multiply. Write the answer in standard form.
Question 37.
3i(−5 + i)
Answer:

Question 38.
2i(7 − i)
Answer:

Question 39.
(3 − 2i)(4 + i)
Answer:

Question 40.
(7 + 5i)(8 − 6i)
Answer:

Question 41.
(4 − 2i)(4 + 2i)
Answer:

Question 42.
(9 + 5i)(9 − 5i)
Answer:

Question 43.
(3 − 6i)²
Answer:

Question 44.
(8 + 3i)²
Answer:

JUSTIFYING STEPS In Exercises 45 and 46, justify each step in performing the operation.
Question 45.
11 − (4 + 3i) + 5i

Answer:

Question 46.
(3 + 2i)(7 − 4i)

Answer:

REASONING In Exercises 47 and 48, place the tiles in the expression to make a true statement.
Question 47.
(____ − ____i) – (____ − ____i ) = 2 − 4i

Answer:

Question 48.
____i(____ + ____i ) = −18 − 10i

Answer:

In Exercises 49–54, solve the equation. Check your solution(s).
Question 49.
x² + 9 = 0
Answer:

Question 50.
x² + 49 = 0
Answer:

Question 51.
x² − 4 = −11
Answer:

Question 52.
x² − 9 = −15
Answer:

Question 53.
2x² + 6 = −34
Answer:

Question 54.
x² + 7 = −47
Answer:

In Exercises 55–62, find the zeros of the function.
Question 55.
f(x) = 3x² + 6
Answer:

Question 56.
g(x) = 7x² + 21
Answer:

Question 57.
h(x) = 2x² + 72
Answer:

Question 58.
k(x) = −5x² − 125
Answer:

Question 59.
m(x) = −x² − 27
Answer:

Question 60.
p(x) = x² + 98
Answer:

Question 61.
r(x) = − \(\frac{1}{2}\)x² − 24
Answer:

Question 62.
f(x) = −\(\frac{1}{5}\)x² − 10
Answer:

ERROR ANALYSIS In Exercises 63 and 64, describe and correct the error in performing the operation and writing the answer in standard form.
Question 63.

Answer:

Question 64.

Answer:

Question 65.
NUMBER SENSE
Simplify each expression. Then classify your results in the table below.
a. (−4 + 7i) + (−4 − 7i)
b. (2 − 6i) − (−10 + 4i)
c. (25 + 15i) − (25 − 6i)
d. (5 + i)(8 − i)
e. (17 − 3i) + (−17 − 6i)
f. (−1 + 2i)(11 − i)
g. (7 + 5i) + (7 − 5i)
h. (−3 + 6i) − (−3 − 8i)

Answer:

Question 66.
MAKING AN ARGUMENT
The Product Property ofSquare Roots states \(\sqrt{a}\) • \(\sqrt{b}\) = \(\sqrt{ab}\) . Your friend concludes \(\sqrt{-4}\) • \(\sqrt{-9}\) = \(\sqrt{36}\) = 6. Is your friend correct? Explain.
Answer:

Question 67.
FINDING A PATTERN
Make a table that shows the powers of i from i¹ to i⁸ in the first row and the simplified forms of these powers in the second row. Describe the pattern you observe in the table. Verify the pattern continues by evaluating the next four powers of i.
Answer:

Question 68.
HOW DO YOU SEE IT?
The graphs of three functions are shown. Which function(s) has real zeros? imaginary zeros? Explain your reasoning.

Answer:

In Exercises 69–74, write the expression as a complex number in standard form.
Question 69.
(3 + 4i) − (7 − 5i) + 2i(9 + 12i)
Answer:

Question 70.
3i(2 + 5i) + (6 − 7i) − (9 + i)
Answer:

Question 71.
(3 + 5i)(2 − 7i⁴)
Answer:

Question 72.
2i³(5 − 12i )
Answer:

Question 73.
(2 + 4i⁵) + (1 − 9i⁶) − (3 +i⁷)
Answer:

Question 74.
(8 − 2i⁴) + (3 − 7i⁸) − (4 + i⁹)
Answer:

Question 75.
OPEN-ENDED
Find two imaginary numbers whose sum and product are real numbers. How are the imaginary numbers related?
Answer:

Question 76.
COMPARING METHODS
Describe the two different methods shown for writing the complex expression in standard form. Which method do you prefer? Explain.

Answer:

Question 77.
CRITICAL THINKING
Determine whether each statement is true or false. If it is true, give an example. If it is false, give a counterexample.
a. The sum of two imaginary numbers is an imaginary number.
b. The product of two pure imaginary numbers is a real number.
c. A pure imaginary number is an imaginary number.
d. A complex number is a real number.
Answer:

Question 78.
THOUGHT PROVOKING
Create a circuit that has an impedance of 14 − 3i.
Answer:

Maintaining Mathematical Proficiency

Determine whether the given value of x is a solution to the equation.
Question 79.
3(x − 2) + 4x − 1 = x − 1; x = 1
Answer:

Question 80.
x³ − 6 = 2x² + 9 − 3x; x = −5
Answer:

Question 81.
−x² + 4x = 19 — 3x²; x = −\(\frac{3}{4}\)
Answer:

Write a quadratic function in vertex form whose graph is shown.
Question 82.

Answer:

Question 83.

Answer:

Question 84.

Answer:

Lesson 3.3 Completing the Square

Essential Question How can you complete the square for a quadratic expression?

EXPLORATION 1

Using Algebra Tiles to Complete the Square
Work with a partner. Use algebra tiles to complete the square for the expression x² + 6x.

a. You can model x² + 6x using one x²-tile and six x-tiles. Arrange the tiles in a square. Your arrangement will be incomplete in one of the corners.
b. How many 1-tiles do you need to complete the square?
c. Find the value of c so that the expression x² + 6x + c is a perfect square trinomial.
d. Write the expression in part (c) as the square of a binomial.

EXPLORATION 2

Drawing Conclusions
Work with a partner.
a. Use the method outlined in Exploration 1 to complete the table.

b. Look for patterns in the last column of the table. Consider the general statement x² + bx + c = (x + d)². How are d and b related in each case? How are c and d related in each case?
c. How can you obtain the values in the second column directly from the coefficients of x in the first column?

Communicate Your Answer

Question 3.
How can you complete the square for a quadratic expression?
Answer:

Question 4.
Describe how you can solve the quadratic equation x² + 6x = 1 by completing the square.
Answer:

Monitoring Progress

Solve the equation using square roots. Check your solution(s).
Question 1.
x² + 4x + 4 = 36
Answer:

Question 2.
x² − 6x + 9 = 1
Answer:

Question 3.
x² − 22x + 121 = 81
Answer:

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
Question 4.
x² + 8x + c
Answer:

Question 5.
x² − 2x + c
Answer:

Question 6.
x² − 9x + c
Answer:

Solve the equation by completing the square.
Question 7.
x² − 4x + 8 = 0
Answer:

Question 8.
x² + 8x − 5 = 0
Answer:

Question 9.
−3x² − 18x − 6 = 0
Answer:

Question 10.
4x² + 32x = −68
Answer:

Question 11.
6x(x + 2) = −42
Answer:

Question 12.
2x(x − 2) = 200
Answer:

Write the quadratic function in vertex form. Then identify the vertex.
Question 13.
y = x² − 8x + 18
Answer:

Question 14.
y = x² + 6x + 4
Answer:

Question 15.
y = x² − 2x − 6
Answer:

Question 16.
WHAT IF?
The height of the baseball can be modeled by y = −16t² + 80t + 2. Find the maximum height of the baseball. How long does the ball take to hit the ground?
Answer:

Completing the Square 3.3 Exercises

Vocabulary and Core Concept Check
Question 1.
VOCABULARY
What must you add to the expression x² + bx to complete the square?
Answer:

Question 2.
COMPLETE THE SENTENCE
The trinomial x² − 6x + 9 is a ____ because it equals ____.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–10, solve the equation using square roots. Check your solution(s).
Question 3.
x² − 8x + 16 = 25
Answer:

Question 4.
r² − 10r + 25 = 1
Answer:

Question 5.
x² − 18x + 81 = 5
Answer:

Question 6.
m² + 8m + 16 = 45
Answer:

Question 7.
y² − 24y + 144 = −100
Answer:

Question 8.
x² − 26x + 169 = −13
Answer:

Question 9.
4w² + 4w + 1 = 75
Answer:

Question 10.
4x² − 8x + 4 = 1
Answer:

In Exercises 11–20, find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
Question 11.
x² + 10x + c
Answer:

Question 12.
x² + 20x + c
Answer:

Question 13.
y² − 12y + c
Answer:

Question 14.
t² − 22t + c
Answer:

Question 15.
x² − 6x + c
Answer:

Question 16.
x² + 24x + c
Answer:

Question 17.
z² − 5z + c
Answer:

Question 18.
x² + 9x + c
Answer:

Question 19.
w² + 13w + c
Answer:

Question 20.
s² − 26s + c
Answer:

In Exercises 21–24, find the value of c. Then write an expression represented by the diagram.
Question 21.

Answer:

Question 22.

Answer:

Question 23.

Answer:

Question 24.

Answer:

In Exercises 25–36, solve the equation by completing the square.
Question 25.
x² + 6x + 3 = 0
Answer:

Question 26.
s² + 2s − 6 = 0
Answer:

Question 27.
x² + 4x − 2 = 0
Answer:

Question 28.
t² − 8t − 5 = 0
Answer:

Question 29.
z(z + 9) = 1
Answer:

Question 30.
x(x + 8) = −20
Answer:

Question 31.
7t² + 28t + 56 = 0
Answer:

Question 32.
6r² + 6r + 12 = 0
Answer:

Question 33.
5x(x + 6) = −50
Answer:

Question 34.
4w(w − 3) = 24
Answer:

Question 35.
4x² − 30x = 12 + 10x
Answer:

Question 36.
3s² + 8s = 2s − 9
Answer:

Question 37.
ERROR ANALYSIS
Describe and correct the error in solving the equation.

Answer:

Question 38.
ERROR ANALYSIS
Describe and correct the error in finding the value of c that makes the expression a perfect square trinomial.

Answer:

Question 39.
WRITING
Can you solve an equation by completing the square when the equation has two imaginary solutions? Explain.
Answer:

Question 40.
ABSTRACT REASONING
Which of the following are solutions of the equation x² − 2ax + a² = b²? Justify your answers.
A. ab
B. −a − b
C. b
D. a
E. a − b
F. a + b
Answer:

USING STRUCTURE In Exercises 41–50, determine whether you would use factoring, square roots, or completing the square to solve the equation. Explain your reasoning. Then solve the equation.
Question 41.
x² − 4x − 21 = 0
Answer:

Question 42.
x² + 13x + 22 = 0
Answer:

Question 43.
(x + 4)² = 16
Answer:

Question 44.
(x − 7)² = 9
Answer:

Question 45.
x² + 12x + 36 = 0
Answer:

Question 46.
x² − 16x + 64 = 0
Answer:

Question 47.
2x² + 4x − 3 = 0
Answer:

Question 48.
3x² + 12x + 1 = 0
Answer:

Question 49.
x² − 100 = 0
Answer:

Question 50.
4x² − 20 = 0
Answer:

MATHEMATICAL CONNECTIONS In Exercises 51–54, find the value of x.
Question 51.
Area of rectangle = 50

Answer:

Question 52.
Area of parallelogram = 48

Answer:

Question 53.
Area of triangle = 40

Answer:

Question 54.
Area of trapezoid = 20

Answer:

In Exercises 55–62, write the quadratic function in vertex form. Then identify the vertex.
Question 55.
f(x) = x² − 8x + 19
Answer:

Question 56.
g(x) = x² − 4x − 1
Answer:

Question 57.
g(x) = x² + 12x + 37
Answer:

Question 58.
h(x) = x² + 20x + 90
Answer:

Question 59.
h(x) = x² + 2x − 48
Answer:

Question 60.
f(x) = x² + 6x − 16
Answer:

Question 61.
f(x) = x² − 3x + 4
Answer:

Question 62.
g(x) = x² + 7x + 2
Answer:

Question 63.
MODELING WITH MATHEMATICS
While marching, a drum major tosses a baton into the air and catches it. The height h (in feet) of the baton t seconds after it is thrown can be modeled by the function h = −16t² + 32t + 6.
a. Find the maximum height of the baton.
b. The drum major catches the baton when it is 4 feet above the ground. How long is the baton in the air?
Answer:

Question 64.
MODELING WITH MATHEMATICS
A firework explodes when it reaches its maximum height. The height h (in feet) of the firework t seconds after it is launched can be modeled by h = \(-\frac{500}{9} t^{2}+\frac{1000}{3} t\) + 10. What is the maximum height of the firework? How long is the firework in the air before it explodes?

Answer:

Question 65.
COMPARING METHODS
A skateboard shop sells about 50 skateboards per week when the advertised price is charged. For each $1 decrease in price, one additional skateboard per week is sold. The shop’s revenue can be modeled by y = (70 − x)(50 + x).

a. Use the intercept form of the function to find the maximum weekly revenue.
b. Write the function in vertex form to find the maximum weekly revenue.
c. Which way do you prefer? Explain your reasoning.
Answer:

Question 66.
HOW DO YOU SEE IT?
The graph of the function f(x) = (x − h)² is shown. What is the x-intercept? Explain your reasoning.

Answer:

Question 67.
WRITING
At Buckingham Fountain in Chicago, the height h (in feet) of the water above the main nozzle can be modeled by h = −16² + 89.6t, where t is the time (in seconds) since the water has left the nozzle. Describe three different ways you could find the maximum height the water reaches. Then choose a method and find the maximum height of the water.
Answer:

Question 68.
PROBLEM SOLVING
A farmer is building a rectangular pen along the side of a barn for animals. The barn will serve as one side of the pen. The farmer has 120 feet of fence to enclose an area of 1512 square feet and wants each side of the pen to be at least 20 feet long.
a. Write an equation that represents the area of the pen.
b. Solve the equation in part (a) to find the dimensions of the pen.

Answer:

Question 69.
MAKING AN ARGUMENT
Your friend says the equation x² + 10x = −20 can be solved by either completing the square or factoring. Is your friend correct? Explain.
Answer:

Question 70.
THOUGHT PROVOKING
Write a function g in standard form whose graph has the same x-intercepts as the graph of f(x) = 2x² + 8x + 2. Find the zeros of each function by completing the square. Graph each function.
Answer:

Question 71.
CRITICAL THINKING
Solve x² + bx + c = 0 by completing the square. Your answer will be an expression for x in terms of b and c.
Answer:

Question 72.
DRAWING CONCLUSIONS
In this exercise, you will investigate the graphical effect of completing the square.
a. Graph each pair of functions in the same coordinate plane.
y = x² + 2x y = x² − 6x
y = (x + 1)² y = (x − 3)²
b. Compare the graphs of y = x² + bx and y = (x + \(\frac{b}{2}\))². Describe what happens to the graph of y = x² + bx when you complete the square.
Answer:

Question 73.
MODELING WITH MATHEMATICS
In your pottery class, you are given a lump of clay with a volume of 200 cubic centimeters and are asked to make a cylindrical pencil holder. The pencil holder should be 9 centimeters high and have an inner radius of 3 centimeters. What thickness x should your pencil holder have if you want to use all of the clay?

Answer:

Maintaining Mathematical Proficiency

Solve the inequality. Graph the solution.
Question 74.
2x − 3 < 5
Answer:

Question 75.
4 − 8y ≥ 12
Answer:

Question 76.
\(\frac{n}{3}\) + 6 > 1
Answer:

Question 77.
−\(\frac{2s}{5}\) ≤ 8
Answer:

Graph the function. Label the vertex, axis of symmetry, and x-intercepts.
Question 78.
g(x) = 6(x − 4)²
Answer:

Question 79.
h(x) = 2x(x − 3)
Answer:

Question 80.
f(x) = x² + 2x + 5
Answer:

Question 81.
f(x) = 2(x + 10)(x − 12)
Answer:

Quadratic Equations and Complex Numbers Study Skills: Creating a Positive Study Environment

3.1–3.3 What Did You Learn?

Core Vocabulary

Core Concepts

Mathematical Practices
Question 1.
Analyze the givens, constraints, relationships, and goals in Exercise 61 on page 101.
Answer:

Question 2.
Determine whether it would be easier to find the zeros of the function in Exercise 63 on page 117 or Exercise 67 on page 118.
Answer:

Study Skills: Creating a Positive Study Environment

• Set aside an appropriate amount of time for reviewing your notes and the textbook, reworking your notes, and completing homework.
• Set up a place for studying at home that is comfortable, but not too comfortable. The place needs to be away from all potential distractions.
• Form a study group. Choose students who study well together, help out when someone misses school, and encourage positive attitudes.
  

Quadratic Equations and Complex Numbers 3.1–3.3 Quiz

Solve the equation by using the graph. Check your solution(s).
Question 1.
x² − 10x + 25 = 0

Answer:

Question 2.
2x² + 16 = 12x

Answer:

Question 3.
x² = −2x + 8

Answer:

Solve the equation using square roots or by factoring. Explain the reason for your choice.
Question 4.
2x² − 15 = 0
Answer:

Question 5.
3x² − x − 2 = 0
Answer:

Question 6.
(x + 3)² = 8
ans;

Question 7.
Find the values of x and y that satisfy the equation 7x − 6i = 14 + yi.
Answer:

Perform the operation. Write your answer in standard form
Question 8.
(2 + 5i) + (−4 + 3i)
Answer:

Question 9.
(3 + 9i) − (1 − 7i)
Answer:

Question 10.
(2 + 4i)(−3 − 5i)
Answer:

Question 11.
Find the zeros of the function f(x) = 9x² + 2. Does the graph of the function intersect the x-axis? Explain your reasoning.
Answer:

Solve the equation by completing the square.
Question 12.
x² − 6x + 10 = 0
Answer:

Question 13.
x² + 12x + 4 = 0
Answer:

Question 14.
4x(x + 6) = −40
Answer:

Question 15.
Write y = x² − 10x + 4 in vertex form. Then identify the vertex.
Answer:

Question 16.
A museum has a café with a rectangular patio. The museum wants to add 464 square feet to the area of the patio by expanding the existing patio as shown.

a. Find the area of the existing patio.
b. Write an equation to model the area of the new patio.
c. By what distance x should the length of the patio be expanded?
Answer:

Question 17.
Find the impedance of the series circuit.

Answer:

Question 18.
The height h (in feet) of a badminton birdie t seconds after it is hit can be modeled by the function h = −16t² + 32t + 4.
a. Find the maximum height of the birdie.
b. How long is the birdie in the air?
Answer:

Lesson 3.4 Using the Quadratic Formula

Essential Question How can you derive a general formula for solving a quadratic equation?

EXPLORATION 1

Deriving the Quadratic Formula
Work with a partner. Analyze and describe what is done in each step in the development of the Quadratic Formula.

EXPLORATION 2

Using the Quadratic Formula
Work with a partner. Use the Quadratic Formula to solve each equation.
a. x² − 4x + 3 = 0
b. x² − 2x + 2 = 0
c. x² + 2x − 3 = 0
d. x² + 4x + 4 = 0
e. x² − 6x + 10 = 0
f. x² + 4x + 6 = 0

Communicate Your Answer

Question 3.
How can you derive a general formula for solving a quadratic equation?
Answer:

Question 4.
Summarize the following methods you have learned for solving quadratic equations: graphing, using square roots, factoring, completing the square, and using the Quadratic Formula.
Answer:

Monitoring Progress

Solve the equation using the Quadratic Formula.
Question 1.
x² − 6x + 4 = 0
Answer:

Question 2.
2x² + 4 = −7x
Answer:

Question 3.
5x² = x + 8
Answer:

Solve the equation using the Quadratic Formula.
Question 4.
x² + 41 = −8x
Answer:

Question 5.
−9x² = 30x + 25
Answer:

Question 6.
5x − 7x² = 3x + 4
Answer:

Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation.
Question 7.
4x² + 8x + 4 = 0
Answer:

qm 8.
\(\frac{1}{2}\)x² + x − 1 = 0
Answer:

Question 9.
5x² = 8x − 13
Answer:

Question 10.
7x² − 3x = 6
Answer:

Question 11.
4x² + 6x = −9
Answer:

Question 12.
−5x² + 1 = 6 − 10x
Answer:

Question 13.
Find a possible pair of integer values for a and c so that the equation ax² + 3x + c = 0 has two real solutions. Then write the equation.
Answer:

Question 14.
WHAT IF?
The ball leaves the juggler’s hand with an initial vertical velocity of 40 feet per second. How long is the ball in the air?
Answer:

Using the Quadratic Formula 3.4 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
When a, b, and c are real numbers such that a ≠ 0, the solutions of the quadratic equation ax² + bx + c = 0 are x= ____________.
Answer:

Question 2.
COMPLETE THE SENTENCE
You can use the ____________ of a quadratic equation to determine the number and type of solutions of the equation.
Answer:

Question 3.
WRITING
Describe the number and type of solutions when the value of the discriminant is negative.
Answer:

Question 4.
WRITING
Which two methods can you use to solve any quadratic equation? Explain when you might prefer to use one method over the other.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–18, solve the equation using the Quadratic Formula. Use a graphing calculator to check your solution(s).
Question 5.
x² − 4x + 3 = 0
Answer:

Question 6.
3x² + 6x + 3 = 0
Answer:

Question 7.
x² + 6x + 15 = 0
Answer:

Question 8.
6x² − 2x + 1 = 0
Answer:

Question 9.
x² − 14x = −49
Answer:

Question 10.
2x² + 4x = 30
Answer:

Question 11.
3x² + 5 = −2x
Answer:

Question 12.
−3x = 2x² − 4
Answer:

Question 13.
−10x = −25 − x²
Answer:

Question 14.
−5x² − 6 = −4x
Answer:

Question 15.
−4x² + 3x = −5
Answer:

Question 16.
x² + 121 = −22x
Answer:

Question 17.
−z² = −12z + 6
Answer:

Question 18.
−7w + 6 = −4w²
Answer:

In Exercises 19–26, find the discriminant of the quadratic equation and describe the number and type of solutions of the equation.
Question 19.
x² + 12x + 36 = 0
Answer:

Question 20.
x² − x + 6 = 0
Answer:

Question 21.
4n² − 4n − 24 = 0
Answer:

Question 22.
−x² + 2x + 12 = 0
Answer:

Question 23.
4x² = 5x − 10
Answer:

Question 24.
−18p = p² + 81
Answer:

Question 25.
24x = −48 − 3x²
Answer:

Question 26.
−2x² − 6 = x²
Answer:

Question 27.
USING EQUATIONS
What are the complex solutions of the equation 2x²− 16x+ 50 = 0?
A. 4 + 3i, 4 − 3i
B. 4 + 12i, 4 − 12i
C. 16 + 3i, 16 − 3i
D. 16 + 12i, 16 − 12i
Answer:

Question 28.
USING EQUATIONS
Determine the number and type of solutions to the equation x² + 7x = −11.
A. two real solutions
B. one real solution
C. two imaginary solutions
D. one imaginary solution
Answer:

ANALYZING EQUATIONS In Exercises 29–32, use the discriminant to match each quadratic equation with the correct graph of the related function. Explain your reasoning.
Question 29.
x² − 6x + 25 = 0
Answer:

Question 30.
2x² − 20x + 50 = 0
Answer:

Question 31.
3x² + 6x − 9 = 0
Answer:

Question 32.
5x² − 10x − 35 = 0
Answer:

ERROR ANALYSIS In Exercises 33 and 34, describe and correct the error in solving the equation.
Question 33.

Answer:

Question 34.

Answer:

OPEN-ENDED In Exercises 35–40, find a possible pair of integer values for a and c so that the quadratic equation has the given solution(s). Then write the equation.
Question 35.
ax² + 4x + c = 0; two imaginary solutions
Answer:

Question 36.
ax² + 6x + c = 0; two real solutions
Answer:

Question 37.
ax² − 8x + c = 0; two real solutions
Answer:

Question 38.
ax² − 6x + c = 0; one real solution
Answer:

Question 39.
ax² + 10x = c; one real solution
Answer:

Question 40.
−4x + c = −ax²; two imaginary solutions
Answer:

USING STRUCTURE In Exercises 41–46, use the Quadratic Formula to write a quadratic equation that has the given solutions.
Question 41.
x = \(\frac{-8 \pm \sqrt{-176}}{-10}\)
Answer:

Question 42.
x = \(\frac{15 \pm \sqrt{-215}}{22}\)
Answer:

Question 43.
x = \(\frac{-4 \pm \sqrt{-124}}{-14}\)
Answer:

Question 44.
x = \(\frac{-9 \pm \sqrt{137}}{4}\)
Answer:

Question 45.
x = \(\frac{-4 \pm 2}{6}\)
Answer:

Question 46.
x = \(\frac{2 \pm 4}{-2}\)
Answer:

COMPARING METHODS In Exercises 47–58, solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain.
Question 47.
3x² − 21 = 3
Answer:

Question 48.
5x² + 38 = 3
Answer:

Question 49.
2x² − 54 = 12x
Answer:

Question 50.
x² = 3x + 15
Answer:

Question 51.
x² − 7x + 12 = 0
Answer:

Question 52.
x² + 8x − 13 = 0
Answer:

Question 53.
5x² − 50x = −135
Answer:

Question 54.
8x² + 4x + 5 = 0
Answer:

Question 55.
−3 = 4x² + 9x
Answer:

Question 56.
−31x + 56 = −x²
Answer:

Question 57.
x² = 1 − x
Answer:

Question 58.
9x² + 36x + 72 = 0
Answer:

MATHEMATICAL CONNECTIONS In Exercises 59 and 60, find the value for x.
Question 59.
Area of the rectangle = 24 m²

Answer:

Question 6.
Area of the triangle = 8ft²

Answer:

Question 61.
MODELING WITH MATHEMATICS
A lacrosse player throws a ball in the air from an initial height of 7 feet. The ball has an initial vertical velocity of 90 feet per second. Another player catches the ball when it is 3 feet above the ground. How long is the ball in the air?

Answer:

Question 62.
NUMBER SENSE
Suppose the quadratic equation ax² + 5x + c = 0 has one real solution. Is it possible for a and c to be integers? rational numbers? Explain your reasoning. Then describe the possible values of a and c.
Answer:

Question 63.
MODELING WITH MATHEMATICS
In a volleyball game, a player on one team spikes the ball over the net when the ball is 10 feet above the court. The spike drives the ball downward with an initial vertical velocity of 55 feet per second. How much time does the opposing team have to return the ball before it touches the court?
Answer:

Question 64.
MODELING WITH MATHEMATICS
An archer is shooting at targets. The height of the arrow is 5 feet above the ground. Due to safety rules, the archer must aim the arrow parallel to the ground.

a. How long does it take for the arrow to hit a target that is 3 feet above the ground?
b. What method did you use to solve the quadratic equation? Explain.
Answer:

Question 65.
PROBLEM SOLVING
A rocketry club is launching model rockets. The launching pad is 30 feet above the ground. Your model rocket has an initial vertical velocity of 105 feet per second. Your friend’s model rocket has an initial vertical velocity of 100 feet per second.
a. Use a graphing calculator to graph the equations of both model rockets. Compare the paths.
b. After how many seconds is your rocket 119 feet above the ground? Explain the reasonableness of your answer(s).
Answer:

Question 66.
PROBLEM SOLVING
The number A of tablet computers sold (in millions) can be modeled by the function A = 4.5t² + 43.5t + 17, where t represents the year after 2010.

a. In what year did the tablet computer sales reach 65 million?
b. Find the average rate of change from 2010 to 2012 and interpret the meaning in the context of the situation.
c. Do you think this model will be accurate after a new, innovative computer is developed? Explain.
Answer:

Question 67.
MODELING WITH MATHEMATICS
A gannet is a bird that feeds on fish by diving into the water. A gannet spots a fish on the surface of the water and dives 100 feet to catch it. The bird plunges toward the water with an initial vertical velocity of −88 feet per second.

a. How much time does the fish have to swim away?
b. Another gannet spots the same fish, and it is only 84 feet above the water and has an initial vertical velocity of −70 feet per second. Which bird will reach the fish first? Justify your answer.
Answer:

Question 68.
USING TOOLS
You are asked to find a possible pair of integer values for a and c so that the equation ax² − 3x + c = 0 has two real solutions. When you solve the inequality for the discriminant, you obtain ac < 2.25. So, you choose the values a = 2 and c = 1. Your graphing calculator displays the graph of your equation in a standard viewing window. Is your solution correct? Explain.

Answer:

Question 69.
PROBLEM SOLVING
Your family has a rectangular pool that measures 18 feet by 9 feet. Your family wants to put a deck around the pool but is not sure how wide to make the deck. Determine how wide the deck should be when the total area of the pool and deck is 400 square feet. What is the width of the deck?

Answer:

Question 70.
HOW DO YOU SEE IT?
The graph of a quadratic function y = ax² + bx + c is shown. Determine whether each discriminant of ax² + bx + c = 0 is positive, negative, or zero. Then state the number and type of solutions for each graph. Explain your reasoning.

Answer:

Question 71.
CRITICAL THINKING
Solve each absolute value equation.
a. |x² – 3x – 14| = 4
b. x² = |x| + 6
Answer:

Question 72.
MAKING AN ARGUMENT
The class is asked to solve the equation 4x² + 14x + 11 = 0. You decide to solve the equation by completing the square. Your friend decides to use the Quadratic Formula. Whose method is more efficient? Explain your reasoning.
Answer:

Question 73.
ABSTRACT REASONING
For a quadratic equation ax² + bx + c = 0 with two real solutions, show that the mean of the solutions is \(\frac{b}{2a}\). How is this fact related to the symmetry of the graph of y = ax² + bx + c?
Answer:

Question 74.
THOUGHT PROVOKING
Describe a real-life story that could be modeled by h = −16t² + v₀t + h₀ . Write the height model for your story and determine how long your object is in the air.
Answer:

Question 75.
REASONING
Show there is no quadratic equation ax²+bx+c= 0 such that a, b, and c are real numbers and 3i and −2i are solutions.
Answer:

Question 76.
MODELING WITH MATHEMATICS
The Stratosphere Tower in Las Vegas is 921 feet tall and has a “needle” at its top that extends even higher into the air. A thrill ride called Big Shot catapults riders 160 feet up the needle and then lets them fall back to the launching pad.

a. The height h (in feet) of a rider on the Big Shot can be modeled by h = −16t² + v₀ t + 921, where t is the elapsed time (in seconds) after launch and v₀ is the initial vertical velocity (in feet per second). Find v0 using the fact that the maximum value of h is 921 + 160 = 1081 feet.
b. A brochure for the Big Shot states that the ride up the needle takes 2 seconds. Compare this time to the time given by the model h = −16t² + v₀ t + 921, where v0 is the value you found in part (a). Discuss the accuracy of the model.
Answer:

Maintaining Mathematical Proficiency

Solve the system of linear equations by graphing.
Question 77.
−x + 2y = 6
x + 4y = 24
Answer:

Question 78.
y = 2x − 1
y = x + 1
Answer:

Question 79.
3x + y = 4
6x + 2y = −4
Answer:

Question 80.
y = −x + 2
−5x + 5y = 10
Answer:

Graph the quadratic equation. Label the vertex and axis of symmetry.
Question 81.
y = −x² + 2x + 1
Answer:

Question 82.
y = 2x² − x + 3
Answer:

Question 83.
y = 0.5x² + 2x + 5
Answer:

Question 84.
y = −3x² − 2
Answer:

Lesson 3.5 Solving Nonlinear Systems

Essential Question How can you solve a nonlinear system of equations?

EXPLORATION 1

Solving Nonlinear Systems of Equations
Work with a partner. Match each system with its graph. Explain your reasoning. Then solve each system using the graph.

EXPLORATION 2

Solving Nonlinear Systems of Equations
Work with a partner. Look back at the nonlinear system in Exploration 1(f). Suppose you want a more accurate way to solve the system than using a graphical approach.
a. Show how you could use a numerical approach by creating a table. For instance, you might use a spreadsheet to solve the system.

b. Show how you could use an analytical approach. For instance, you might try solving the system by substitution or elimination.

Communicate Your Answer

Question 3.
How can you solve a nonlinear system of equations?
Answer:

Question 4.
Would you prefer to use a graphical, numerical, or analytical approach to solve the given nonlinear system of equations? Explain your reasoning.
Answer:

Solve the system using any method. Explain your choice of method.
Question 1.
y = −x² + 4
y = −4x + 8
Answer:

Question 2.
x² + 3x + y = 0
2x + y = 5
Answer:

Question 3.
2x² + 4x − y =−2
x² + y = 2
Answer:

Solve the system.
Question 4.
x² + y² = 16
y = −x + 4
Answer:

Question 5.
x² + y² = 4
y = x + 4
Answer:

Question 6.
x² + y² = 1
y = \(\frac{1}{2}\)x + \(\frac{1}{2}\)
Answer:

Solve the equation by graphing.
Question 7.
x² − 6x + 15 = −(x − 3)² + 6
Answer:

Question 8.
(x + 4)(x − 1) = −x² + 3x + 4
Answer:

Solving Nonlinear Systems 3.5 Exercises

Vocabulary and Core Concept Check
Question 1.
WRITING
Describe the possible solutions of a system consisting of two quadratic equations.
Answer:

Question 2.
WHICH ONE DOESN’T BELONG?
Which system does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–10, solve the system by graphing. Check your solution(s).
Question 3.
y = x + 2
y = 0.5(x + 2)²
Answer:

Question 4.
y = (x − 3)² + 5
y = 5
Answer:

Question 5.
y = \(\frac{1}{3}\)x + 2
y = −3x² − 5x − 4
Answer:

Question 6.
y = −3x² − 30x − 71
y = −3x − 17
Answer:

Question 7.
y = x² + 8x + 18
y = −2x² − 16x − 30
Answer:

Question 8.
y = −2x² − 9
y = −4x − 1
Answer:

Question 9.
y = (x − 2)²
y = −x² + 4x − 2
Answer:

Question 10.
y = \(\frac{1}{2}\)(x + 2)²
y = −\(\frac{1}{2}\)x² + 2
Answer:

In Exercises 11–14, solve the system of nonlinear equations using the graph.
Question 11.

Answer:

Question 12.

Answer:

Question 13.

Answer:

Question 14.

Answer:

In Exercises 15–24, solve the system by substitution.
Question 15.
y = x + 5
y = x² − x + 2
Answer:

Question 16.
x² + y² = 49
y = 7 − x
Answer:

Question 17.
x² + y² = 64
y = −8
Answer:

Question 18.
x = 3
−3x² + 4x − y = 8
Answer:

Question 19.
2x² + 4x − y = −3
−2x + y = −4
Answer:

Question 20.
2x − 3 = y + 5x²
y = −3x − 3
Answer:

Question 21.
y = x² − 1
−7 = −x² − y
Answer:

Question 22.
y + 16x − 22 = 4x²
4x² − 24x + 26 + y = 0
Answer:

Question 23.
x² + y² = 7
x + 3y = 21
Answer:

Question 24.
x² + y² = 5
−x + y = −1
Answer:

Question 25.
USING EQUATIONS
Which ordered pairs are solutions of the nonlinear system?
y = \(\frac{1}{2}\)x² − 5x + \(\frac{21}{2}\)
y = −\(\frac{1}{2}\)x + \(\frac{13}{2}\)
A. (1, 6)
B. (3, 0)
C. (8, 2.5)
D. (7, 0)
Answer:

Question 26.
USING EQUATIONS
How many solutions does the system have? Explain your reasoning.
y = 7x² − 11x + 9
y = −7x² + 5x − 3
A. 0
B. 1
C. 2
D. 4
Answer:

In Exercises 27–34, solve the system by elimination.
Question 27.
2x² − 3x −y =−5
−x + y = 5
Answer:

Question 28.
−3x² + 2x − 5 = y
−x + 2 = −y
Answer:

Question 29.
−3x² + y = −18x + 29
−3x² − y = 18x − 25
Answer:

Question 30.
y = −x² − 6x 10
y = 3x² + 18x + 22
Answer:

Question 31.
y + 2x = −14
−x² − y − 6x = 11
Answer:

Question 32.
y = x² + 4x + 7
−y = 4x + 7
Answer:

Question 33.
y = −3x² − 30x − 76
y = 2x² + 20x + 44
Answer:

Question 34.
−10x² + y = −80x + 155
5x² + y = 40x − 85
Answer:

Question 35.
ERROR ANALYSIS
Describe and correct the error in using elimination to solve a system.

Answer:

Question 36.
NUMBER SENSE
The table shows the inputs and outputs of two quadratic equations. Identify the solution(s) of the system. Explain your reasoning.

Answer:

In Exercises 37–42, solve the system using any method. Explain your choice of method.
Question 37.
y = x² − 1
−y = 2x² + 1
Answer:

Question 38.
y = −4x² − 16x − 13
−3x² + y + 12x = 17
Answer:

Question 39.
−2x + 10 + y = \(\frac{1}{3}\)x²
y = 10
Answer:

Question 40.
y = 0.5x² − 10
y = −x² + 14
Answer:

Question 41.
y = −3(x − 4)² + 6
(x − 4)² + 2 − y = 0
Answer:

Question 42.
−x² + y² = 100
y = −x + 14
Answer:

USING TOOLS In Exercises 43–48, solve the equation by graphing.
Question 43.
x² + 2x = −\(\frac{1}{2}\)x² + 2x
Answer:

Question 44.
2x² − 12x − 16 = −6x² + 60x − 144
Answer:

Question 45.
(x + 2)(x − 2) = −x² + 6x − 7
Answer:

Question 46.
−2x² − 16x − 25 = 6x² + 48x + 95
Answer:

Question 47.
(x − 2)² − 3 = (x + 3)(−x + 9) − 38
Answer:

Question 48.
(−x + 4)(x + 8) − 42 = (x + 3)(x + 1) − 1
Answer:

Question 49.
REASONING
A nonlinear system contains the equations of a constant function and a quadratic function. The system has one solution. Describe the relationship between the graphs.
Answer:

Question 50.
PROBLEM SOLVING
The range (in miles) of a broadcast signal from a radio tower is bounded by a circle given by the equation x² + y²= 1620.

A straight highway can be modeled by the equation y = −\(\frac{1}{3}\)x + 30.
For what lengths of the highway are cars able to receive the broadcast signal?
Answer:

Question 51.
PROBLEM SOLVING
A car passes a parked police car and continues at a constant speed r. The police car begins accelerating at a constant rate when it is passed. The diagram indicates the distance d (in miles) the police car travels as a function of time t (in minutes) after being passed. Write and solve a system of equations to find how long it takes the police car to catch up to the other car.

Answer:

Question 52.
THOUGHT PROVOKING
Write a nonlinear system that has two different solutions with the same y-coordinate. Sketch a graph of your system. Then solve the system.
Answer:

Question 53.
OPEN-ENDED
Find three values for m so the system has no solution, one solution, and two solutions. Justify your answer using a graph.
3y = −x² + 8x − 7
y = mx + 3
Answer:

Question 54.
MAKING AN ARGUMENT
You and a friend solve the system shown and determine that x = 3 and x = −3. You use Equation 1 to obtain the solutions (3, -3), (3, −3), (−3, 3), and (−3, −3). Your friend uses Equation 2 to obtain the solutions (3, 3) and (−3, −3). Who is correct? Explain your reasoning.
x² + y² = 18 Equation 1
x − y = 0 Equation 2
Answer:

Question 55.
COMPARING METHODS
Describe two different ways you could solve the quadratic equation. Which way do you prefer? Explain your reasoning.
−2x² + 12x − 17 = 2x² − 16x + 31
Answer:

Question 56.
ANALYZING RELATIONSHIPS
Suppose the graph of a line that passes through the origin intersects the graph of a circle with its center at the origin. When you know one of the points of intersection, explain how you can find the other point of intersection without performing any calculations.
Answer:

Question 57.
WRITING
Describe the possible solutions of a system that contains (a) one quadratic equation and one equation of a circle, and (b) two equations of circles. Sketch graphs to justify your answers.
Answer:

Question 58.
HOW DO YOU SEE IT?
The graph of a nonlinear system is shown. Estimate the solution(s). Then describe the transformation of the graph of the linear function that results in a system with no solution.

Answer:

Question 59.
MODELING WITH MATHEMATICS
To be eligible for a parking pass on a college campus, a student must live at least 1 mile from the campus center.

a. Write equations that represent the circle and Oak Lane.
b. Solve the system that consists of the equations in part (a).
c. For what length of Oak Lane are students not eligible for a parking pass?
Answer:

Question 60.
CRITICAL THINKING
Solve the system of three equations shown.
x² + y² = 4
2y = x² − 2x + 4
y = −x + 2
Answer:

Maintaining Mathematical Proficiency

Solve the inequality. Graph the solution on a number line.
Question 61.
4x − 4 > 8
Answer:

Question 62.
−x + 7 ≤ 4 − 2x
Answer:

Question 63.
−3(x − 4) ≥ 24
Answer:

Write an inequality that represents the graph.
Question 64.

Answer:

Question 65.

Answer:

Question 66.

Answer:

Lesson 3.6 Quadratic Inequalities

Essential Question How can you solve a quadratic inequality?

EXPLORATION 1

Solving a Quadratic Inequality
Work with a partner. The graphing calculator screen shows the graph of f(x) = x² + 2x − 3.

Explain how you can use the graph to solve the inequality x² + 2x − 3 ≤ 0.
Then solve the inequality.

EXPLORATION 2

Solving Quadratic Inequalities
Work with a partner. Match each inequality with the graph of its related quadratic function. Then use the graph to solve the inequality.
a. x² − 3x + 2 > 0
b. x² − 4x + 3 ≤ 0
c. x² − 2x − 3 < 0
d. x² + x − 2 ≥ 0
e. x² − x − 2 < 0
f. x² − 4 > 0

Communicate Your Answer

Question 3.
How can you solve a quadratic inequality?
Answer:

Question 4.
Explain how you can use the graph in Exploration 1 to solve each inequality. Then solve each inequality.
Answer:

Monitoring Progress

Graph the inequality.
Question 1.
y ≥ x² + 2x − 8
Answer:

Question 2.
y ≤ 2x² −x − 1
Answer:

Question 3.
y > −x² + 2x + 4
Answer:

Question 4.
Graph the system of inequalities consisting of y ≤ −x² and y > x² − 3.
Answer:

Solve the inequality.
Question 5.
2x² + 3x ≤ 2
Answer:

Question 6.
−3x² − 4x + 1 < 0
Answer:

Question 7.
2x² + 2 > −5x
Answer:

Question 8.
WHAT IF?
In Example 6, the area must be at least 8500 square feet. Describe the possible lengths of the parking lot.
Answer:

Quadratic Inequalities 3.6 Exercises

Vocabulary and Core Concept Check
Question 1.
WRITING
Compare the graph of a quadratic inequality in one variable to the graph of a quadratic inequality in two variables.
Answer:

Question 2.
WRITING
Explain how to solve x² + 6x − 8 < 0 using algebraic methods and using graphs.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, match the inequality with its graph. Explain your reasoning.
Question 3.
y ≤ x² + 4x + 3
Answer:

Question 4.
y > −x² + 4x − 3
Answer:

Question 5.
y < x² − 4x + 3
Answer:

Question 6.
y ≥ x² + 4x + 3
Answer:

In Exercises 7–14, graph the inequality.
Question 7.
y < −x²
Answer:

Question 8.
y ≥ 4x²
Answer:

Question 9.
y > x² − 9
Answer:

Question 10.
y < x² + 5
Answer:

Question 11.
y ≤ x² + 5x
Answer:

Question 12.
y ≥ −2x² + 9x − 4
Answer:

Question 13.
y > 2(x + 3)² − 1
Answer:

Question 14.
y ≤ (x − \(\frac{1}{2}\))² + \(\frac{5}{2}\)
Answer:

ANALYZING RELATIONSHIPS In Exercises 15 and 16, use the graph to write an inequality in terms of f (x) so point P is a solution.
Question 15.

Answer:

Question 16.

Answer:

ERROR ANALYSIS In Exercises 17 and 18, describe and correct the error in graphing y ≥ x² + 2.
Question 17.

Answer:

Question 18.

Answer:

Question 19.
MODELING WITH MATHEMATICS
A hardwood shelf in a wooden bookcase can safely support a weight W (in pounds) provided W ≤ 115x², where x is the thickness (in inches) of the shelf. Graph the inequality and interpret the solution.
Answer:

Question 20.
MODELING WITH MATHEMATICS
A wire rope can safely support a weight W (in pounds) provided W ≤ 8000d², where d is the diameter (in inches) of the rope. Graph the inequality and interpret the solution.
Answer:

In Exercises 21–26, graph the system of quadratic inequalities.
Question 21.
y ≥ 2x²
y < −x² + 1
Answer:

Question 22.
y > −5x²
y > 3x² − 2
Answer:

Question 23.
y ≤ −x² + 4x − 4
y < x² + 2x − 8
Answer:

Question 24.
y ≥ x² − 4
y ≤ −2x² + 7x + 4
Answer:

Question 25.
y ≥ 2x² + x − 5
y < −x² + 5x + 10
Answer:

Question 26.
y ≥ x² − 3x − 6
y ≥ x² + 7x + 6
Answer:

In Exercises 27–34, solve the inequality algebraically.
Question 27.
4x² < 25
Answer:

Question 28.
x² + 10x + 9 < 0
Answer:

Question 29.
x² − 11x ≥ −28
Answer:

Question 30.
3x² − 13x > −10
Answer:

Question 31.
2x² − 5x − 3 ≤ 0
Answer:

Question 32.
4x² + 8x − 21 ≥ 0
Answer:

Question 33.
\(\frac{1}{2}\)x² − x > 4
Answer:

Question 34.
−\(\frac{1}{2}\)x² + 4x ≤ 1
Answer:

In Exercises 35–42, solve the inequality by graphing.
Question 35.
x² − 3x + 1 < 0
Answer:

Question 36.
x² − 4x + 2 > 0
Answer:

Question 37.
x² + 8x > −7
Answer:

Question 38.
x² + 6x < −3
Answer:

Question 39.
3x² − 8 ≤ − 2x
Answer:

Question 40.
3x² + 5x − 3 < 1
Answer:

Question 41.
\(\frac{1}{3}\)x² + 2x ≥ 2
Answer:

Question 42.
\(\frac{3}{4}\)x² + 4x ≥ 3
Answer:

Question 43.
DRAWING CONCLUSIONS
Consider the graph of the function f(x) = ax² + bx + c.

a. What are the solutions of ax² + bx + c < 0?
b. What are the solutions of ax² + bx + c > 0?
c. The graph of g represents a reflection in the x-axis of the graph of f. For which values of x is g(x) positive?
Answer:

Question 44.
MODELING WITH MATHEMATICS
A rectangular fountain display has a perimeter of 400 feet and an area of at least 9100 feet. Describe the possible widths of the fountain.

Answer:

Question 45.
MODELING WITH MATHEMATICS
The arch of the Sydney Harbor Bridge in Sydney, Australia, can be modeled by y = −0.00211x² + 1.06x, where x is the distance (in meters) from the left pylons and y is the height (in meters) of the arch above the water. For what distances x is the arch above the road?

Answer:

Question 46.
PROBLEM SOLVING
The number T of teams that have participated in a robot-building competition for high-school students over a recent period of time x(in years) can be modeled by T(x) = 17.155x² + 193.68x + 235.81, 0 ≤ x ≤ 6.
After how many years is the number of teams greater than 1000? Justify your answer.

Answer:

Question 47.
PROBLEM SOLVING
A study found that a driver’s reaction time A(x) to audio stimuli and his or her reaction time V(x) to visual stimuli (both in milliseconds) can be modeled by
A(x) = 0.0051x² − 0.319x + 15, 16 ≤ x ≤ 70
V(x) = 0.005x² − 0.23x + 22, 16 ≤ x ≤ 70
where x is the age (in years) of the driver.
a. Write an inequality that you can use to find the x-values for which A(x) is less than V(x).
b. Use a graphing calculator to solve the inequality A(x) < V(x). Describe how you used the domain 16 ≤ x ≤ 70 to determine a reasonable solution. c. Based on your results from parts (a) and (b), do you think a driver would react more quickly to a traffic light changing from green to yellow or to the siren of an approaching ambulance? Explain.
Answer:

Question 48.
HOW DO YOU SEE IT?
The graph shows a system of quadratic inequalities.

a. Identify two solutions of the system.
b. Are the points (1, −2) and (5, 6) solutions of the system? Explain.
c. Is it possible to change the inequality symbol(s) so that one, but not both of the points in part (b), is a solution of the system? Explain.
Answer:

Question 49.
MODELING WITH MATHEMATICS
The length L (in millimeters) of the larvae of the black porgy fish can be modeled by L(x) = 0.00170x² + 0.145x + 2.35, 0 ≤ x ≤ 40 where x is the age (in days) of the larvae. Write and solve an inequality to find at what ages a larva’s length tends to be greater than 10 millimeters. Explain how the given domain affects the solution.

Answer:

Question 50.
MAKING AN ARGUMENT
You claim the system of inequalities below, where a and b are real numbers, has no solution. Your friend claims the system will always have at least one solution. Who is correct? Explain.
y < (x + a)²
y < (x + b)²
Answer:

Question 51.
MATHEMATICAL CONNECTIONS
The area A of the region bounded by a parabola and a horizontal line can be modeled by A= \(\frac{2}{3}\)bh, where b and h are as defined in the diagram. Find the area of the region determined by each pair of inequalities.

Answer:

Question 52.
THOUGHT PROVOKING
Draw a company logo that is created by the intersection of two quadratic inequalities. Justify your answer.
Answer:

Question 53.
REASONING
A truck that is 11 feet tall and 7 feet wide is traveling under an arch. The arch can be modeled by y = −0.0625x² + 1.25x + 5.75, where x and y are measured in feet.

a. Will the truck fit under the arch? Explain.
b. What is the maximum width that a truck 11 feet tall can have and still make it under the arch?
c. What is the maximum height that a truck 7 feet wide can have and still make it under the arch?
Answer:

Maintaining Mathematical Proficiency

Graph the function. Label the x-intercept(s) and the y-intercept.
Question 54.
f(x) = (x + 7)(x − 9)
Answer:

Question 55.
g(x) = (x − 2)² − 4
Answer:

Question 56.
h(x) = −x² + 5x − 6
Answer:

Find the minimum value or maximum value of the function. Then describe where the function is increasing and decreasing.
Question 57.
f(x) = −x² − 6x − 10
Answer:

Question 58.
h(x) = \(\frac{1}{2}\)(x + 2)² − 1
Answer:

Question 59.
f(x) = −(x − 3)(x + 7)
Answer:

Question 60.
h(x) = x² + 3x − 18
Answer:

Quadratic Equations and Complex Numbers Performance Task: Algebra in Genetics: The Hardy-Weinberg Law

3.4–3.6 What Did You Learn?

Core Vocabulary

Core Concepts
Section 3.4
Solving Equations Using the Quadratic Formula, p. 122
Analyzing the Discriminant of ax²+bx+c= 0, p. 124
Methods for Solving Quadratic Equations, p. 125
Modeling Launched Objects, p. 126

Section 3.5
Solving Systems of Nonlinear Equations, p. 132
Solving Equations by Graphing, p. 135

Section 3.6
Graphing a Quadratic Inequality in Two Variables, p. 140
Solving Quadratic Inequalities in One Variable, p. 142

Mathematical Practices
Question 1.
How can you use technology to determine whose rocket lands first in part (b) of Exercise 65 on page 129?
Answer:

Question 2.
What question can you ask to help the person avoid making the error in Exercise 54 on page 138?
Answer:

Question 3.
Explain your plan to find the possible widths of the fountain in Exercise 44 on page 145.
Answer:

Performance Task: Algebra in Genetics: The Hardy-Weinberg Law
Some people have attached earlobes, the recessive trait. Some people have free earlobes, the dominant trait. What percent of people carry both traits?
To explore the answers to this question and more, go to BigIdeasMath.com.

Quadratic Equations and Complex Numbers Chapter Review

3.1 Solving Quadratic Equations (pp. 93–102)

Question 1.
Solve x² − 2x − 8 = 0 by graphing.
Answer:

Solve the equation using square roots or by factoring.
Question 2.
3x² − 4 = 8
Answer:

Question 3.
x² + 6x − 16 = 0
Answer:

Question 4.
2x² − 17x = −30
Answer:

Question 5.
A rectangular enclosure at the zoo is 35 feet long by 18 feet wide. The zoo wants to double the area of the enclosure by adding the same distance x to the length and width. Write and solve an equation to find the value of x. What are the dimensions of the enclosure?
Answer:

3.2 Complex Numbers (pp. 103–110)

Question 6.
Find the values x and y that satisfy the equation 36 − yi = 4x + 3i.
Answer:

Perform the operation. Write the answer in standard form.
Question 7.
(−2 + 3i ) + (7 − 6i )
Answer:

Question 8.
(9 + 3i ) − (−2 − 7i )
Answer:

Question 9.
(5 + 6i )(−4 + 7i )
Answer:

Question 10.
Solve 7x² + 21 = 0.
Answer:

Question 11.
Find the zeros of f(x) = 2x² + 32.
Answer:

3.3 Completing the Square (pp. 111–118)

Question 12.
An employee at a local stadium is launching T-shirts from a T-shirt cannon into the crowd during an intermission of a football game. The height h (in feet) of the T-shirt after t seconds can be modeled by h = −16t² + 96t + 4. Find the maximum height of the T-shirt.
Answer:

Solve the equation by completing the square.
Question 13.
x² + 16x + 17 = 0
Answer:

Question 14.
4x² + 16x + 25 = 0
Answer:

Question 15.
9x(x − 6) = 81
Answer:

Question 16.
Write y = x² − 2x + 20 in vertex form. Then identify the vertex.
Answer:

3.4 Using the Quadratic Formula (pp. 121–130)

Solve the equation using the Quadratic Formula.
Question 17.
−x² + 5x = 2
Answer:

Question 18.
2x² + 5x = 3
Answer:

Question 19.
3x² − 12x + 13 = 0
Answer:

Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation.
Question 20.
−x² − 6x − 9 = 0
Answer:

Question 21.
x² − 2x − 9 = 0
Answer:

Question 22.
x² + 6x + 5 = 0
Answer:

3.5 Solving Nonlinear Systems (pp. 131–138)

Solve the system by any method. Explain your choice of method.
Question 23.
2x² − 2 = y
−2x + 2 = y
Answer:

Question 24.
x² − 6x + 13 = y
−y = −2x + 3
Answer:

Question 25.
x² + y² = 4
−15x + 5 = 5y
Answer:

Question 26.
Solve −3x² + 5x − 1 = 5x² − 8x − 3 by graphing.
Answer:

3.6 Quadratic Inequalities (pp. 139–146)

Graph the inequality.
Question 27.
y > x² + 8x + 16
Answer:

Question 28.
y ≥ x² + 6x + 8
Answer:

Question 29.
x² + y ≤ 7x − 12
Answer:

Graph the system of quadratic inequalities.
Question 30.
x² − 4x + 8 > y
−x² + 4x + 2 ≤ y
Answer:

Question 31.
2x² − x ≥ y − 5
0.5x²> y − 2x− 1
Answer:

Question 32.
−3x² − 2x ≤ y + 1
−2x² + x − 5 > −y
Answer:

Solve the inequality.
Question 33.
3x² + 3x − 60 ≥0
Answer:

Question 34.
−x² − 10x < 21
Answer:

Question 35.
3x² + 2 ≤ 5x
Answer:

Quadratic Equations and Complex Numbers Chapter Test

Solve the equation using any method. Provide a reason for your choice.
Question 1.
0 = x² + 2x + 3
Answer:

Question 2.
6x = x² + 7
Answer:

Question 3.
x² + 49 = 85
Answer:

Question 4.
(x + 4)(x − 1) = −x² + 3x + 4
Answer:

Explain how to use the graph to find the number and type of solutions of the quadratic equation. Justify your answer by using the discriminant.
Question 5.
\(\frac{1}{2}\)x² + 3x + \(\frac{9}{2}\) = 0

Answer:

Question 6.
4x² + 16x + 18 = 0

Answer:

Question 7.
−x² + \(\frac{1}{2}\)x + \(\frac{3}{2}\) = 0

Answer:

Solve the system of equations or inequalities.
Question 8.
x² + 66 = 16x − y
2x − y = 18
Answer:

Question 9.
y ≥ \(\frac{1}{4}\)x² − 2
y < −(x + 3)2x − y = 18 + 4
Answer:

Question 10.
0 = x² + y² − 40
y = x + 4
Answer:

Question 11.
Write (3 + 4i )(4 − 6i ) as a complex number in standard form.
Answer:

Question 12.
The aspect ratio of a widescreen TV is the ratio of the screen’s width to its height, or 16 : 9. What are the width and the height of a 32-inch widescreen TV? Justify your answer. (Hint: Use the Pythagorean Theorem and the fact that TV sizes refer to the diagonal length of the screen.)

Answer:

Question 13.
The shape of the Gateway Arch in St. Louis, Missouri, can be modeled by y = −0.0063x² + 4x, where x is the distance (in feet) from the left foot of the arch and y is the height (in feet) of the arch above the ground. For what distances x is the arch more than 200 feet above the ground? Justify your answer.
Answer:

Question 14.
You are playing a game of horseshoes. One of your tosses is modeled in the diagram, where x is the horseshoe’s horizontal position (in feet) and y is the corresponding height (in feet). Find the maximum height of the horseshoe. Then find the distance the horseshoe travels. Justify your answer.

Answer:

Quadratic Equations and Complex Numbers Cumulative Assessment

Question 1.
The graph of which inequality is shown?

A. y² > x² + x – 6
B. y ≥ x² + x – 6
C. y > x² – x – 6
D. y ≥ x² – x – 6
Answer:

Question 2.
Classify each function by its function family. Then describe the transformation of the parent function.
a. g(x) = x + 4
b. h(x) = 5
c. h(x) = x² − 7
d. g(x) = −∣x + 3∣− 9
e. g(x) = \(\frac{1}{4}\)(x − 2)² − 1
f. h(x) = 6x+ 11
Answer:

Question 3.
Two baseball players hit back-to-back home runs. The path of each home run is modeled by the parabolas below, where x is the horizontal distance (in feet) from home plate and y is the height (in feet) above the ground. Choose the correct symbol for each inequality to model the possible locations of the top of the outfield wall.(HSA-CED.A.3)

Answer:

Question 4.
You claim it is possible to make a function from the given values that has an axis of symmetry at x = 2. Your friend claims it is possible to make a function that has an axis of symmetry at x = −2. What values can you use to support your claim? What values support your friend’s claim?(HSF-IF.B.4)

Answer:

Question 5.
Which of the following values are x-coordinates of the solutions of the system?
y = x² – 6x + 14
y = 2x + 7

Answer:

Question 6.
The table shows the altitudes of a hang glider that descends at a constant rate. How long will it take for the hang glider to descend to an altitude of 100 feet? Justify your answer.

A. 25 seconds
B. 35 seconds
C. 45 seconds
D. 55 seconds
Answer:

Question 7.
Use the numbers and symbols to write the expression x² + 16 as the product of two binomials. Some may be used more than once. Justify your answer.

Answer:

Question 8.
Choose values for the constants h and k in the equation x = \(\frac{1}{4}\)( y − k)² + h so that each statement is true.(HSA-CED.A.2)

Answer:

Question 9.
Which of the following graphs represents a perfect square trinomial? Write each function in vertex form by completing the square.

Answer:

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Big Ideas Math Book 7th Grade Advanced Answer Key | BIM Math 7th Grade Advanced Answers

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• Chapter 1 Equations
• Chapter 2: Transformations
• Chapter 3: Angles and Triangles
• Chapter 4: Graphing and Writing Linear Equations
• Chapter 5: Systems of Linear Equations
• Chapter 6: Data Analysis and Displays
• Chapter 7: Functions
• Chapter 8 Exponents and Scientific Notation
• Chapter 9 Real Numbers and the Pythagorean Theorem
• Chapter 10 Volume and Similar Solids
• Chapter A: Equations and Inequalities
• Chapter B: Probability
• Chapter C: Statistics
• Chapter D: Geometric Shapes and Angles
• Chapter E: Surface Area and Volume

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Big Ideas 4th Grade Math Answers Solutions Pdf | Big Ideas Math Book 4th Grade Answer Key

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• Chapter 1 Place Value Concepts
• Chapter 2 Add and Subtract Multi-Digit Numbers
• Chapter 3 Multiply by One-Digit Numbers
• Chapter 4 Multiply by Two-Digit Numbers
• Chapter 5 Divide Multi-Digit Numbers by One-Digit Numbers
• Chapter 6 Factors, Multiples, and Patterns
• Chapter 7 Understand Fraction Equivalence and Comparison
• Chapter 8 Add and Subtract Fractions
• Chapter 9 Multiply Whole Numbers and Fractions
• Chapter 10 Relate Fractions and Decimals
• Chapter 11 Understand Measurement Equivalence
• Chapter 12 Use Perimeter and Area Formulas
• Chapter 13 Identify and Draw Lines and Angles
• Chapter 14 Identify Symmetry and Two-Dimensional Shapes

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Parabolas describe many natural phenomena like the motion of objects affected by gravity, increase or decrease in the population, amount of reagents in a chemical equation, etc. At times, you need to evaluate how variables change with respect to time. To track how variables change over time, you can put equations into Parametric Form.

Different Parametric Equations can be used to represent a Parabola. We have listed the simple and easiest way on How to find the Parametric Equations of a Parabola in the below modules. Refer to the Solved Examples on Parametric Equations of Parabola for a better understanding of the concept.

Standard Forms of Parabola and their Parametric Equations

Let us discuss in detail the Parametric Coordinates of a Point on Standard Forms of Parabola and their Parametric Equations

Standard Equation of Parabola y² = 4ax

• Parametric Coordinates of the Parabola y² = 4ax are (at², 2at)
• Parametric Equations of Parabola y² = 4ax are x = at² and y = 2at

Standard Equation of Parabola y² = -4ax

• Parametric Coordinates of the Parabola y² = -4ax are (-at², 2at)
• Parametric Equations of Parabola y² = -4ax are x = -at² and y = 2at

Standard Equation of Parabola x² = 4ay

• Parametric Coordinates of the Parabola x² = 4ay are (2at, at²)
• Parametric Equations of Parabola x² = 4ay are x = 2at, y = at²

Standard Equation of Parabola x² = -4ay

• Parametric Coordinates of the Parabola x² = 4ay are (2at, -at²)
• Parametric Equations of Parabola x² = 4ay are x = 2at, y = -at²

Standard Equation of Parabola (y-k)² = 4a(x-h)

Parametric Equations of Parabola (y-k)² = 4a(x-h) are x=h+at², and y = k+2at

Solved Examples on finding the Parametric Equations of a Parabola

1. Write the Parametric Equations of the Parabola y² = 16x?

Solution:

Given Equation is in the form of y² = 4ax

On Comparing the terms we have the 4a = 16

a = 4

The formula for Parametric Equations of the given parabola is x = at² and y = 2at

Substitute the value of a to get the parametric equations i.e. x = 4t² and y = 2*4*t = 8t

Therefore, Parametric Equations of Parabola y² = 16x are x= 4t² and y = 8t

2. Write the Parametric Equations of Parabola x² = 12y?

Solution:

Given Equation is in the form of x² = 4ay

On Comparing the terms we have the 4a = 12

a = 3

The formula for Parametric Equations of the given parabola is x = 2at, and y =  at²

Substitute the value of a to get the parametric equations i.e. x = 2*3*t and y = 3t²

Therefore, Parametric Equations of Parabola x² = 12y are x = 6t and y = 3t²

3. Write the Parametric Equations of the Parabola (y-3)² =8(x-2)?

Solution:

Given Equation is in the form of (y-k)² = 4a(x-2)

Comparing the two equations we have k = 3, h = 2 and 4a = 8 i.e. a =2

The Formula for Parametric Equations of Parabola (y-k)² = 4a(x-h) are x=h+at², and y = k+2at

substitute the values of k, a in the formula and obtain the parametric equation

x = 2+2t² and y = 3+2*2t

x = 2+2t² and y = 3+4t

Therefore, Parametric Equations of the Parabola (y-3)² =8(x-2) are x = 2+2t² and y = 3+4t

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Big Ideas Math 3rd Grade Chapter 4 Division Facts and Strategies Answer Key

By solving the questions from Big Ideas Math Book Grade 3 Chapter 4 Division Facts and Strategies, you will know how to divide two numbers and different methods to solve it. The different topics covered in the Division Facts and Strategies chapter are Use Arrays to Divide, Relate Multiplication and Division, Divide by 2, 5, or 10, Divide by 3 or 4, Divide by 6 or 7, Divide by 8 or 9, and Divide by 0 or 1.

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Lesson – 1: Use Arrays to Divide

• Lesson 4.1 Use Arrays to Divide
• Use Arrays to Divide Homework & Practice 4.1

Lesson – 2: Relate Multiplication and Division

• Lesson 4.2 Relate Multiplication and Division
• Relate Multiplication and Division Homework & Practice 4.2

Lesson – 3: Divide by 2, 5, or 10

• Lesson 4.3 Divide by 2, 5, or 10
• Divide by 2, 5, or 10 Homework & Practice 4.3

Lesson – 4: Divide by 3 or 4

• Lesson 4.4 Divide by 3 or 4
• Divide by 3 or 4 Homework & Practice 4.4

Lesson – 5: Divide by 6 or 7

• Lesson 4.5 Divide by 6 or 7
• Divide by 6 or 7 Homework & Practice 4.5

Lesson – 6: Divide by 8 or 9

• Lesson 4.6 Divide by 8 or 9
• Divide by 8 or 9 Homework & Practice 4.6

Lesson – 7: Divide by 0 or 1

• Lesson 4.7 Divide by 0 or 1
• Divide by 0 or 1 Homework & Practice 4.7

Lesson – 8: Practice Division Strategies

• Lesson 4.8 Practice Division Strategies
• Practice Division Strategies Homework & Practice 4.8

Lesson – 9: Problem Solving: Division

• Lesson 4.9 Problem Solving: Division
• Problem Solving: Division Homework & Practice 4.9

Performance Task

• Division Facts and Strategies Performance Task
• Division Facts and Strategies Activity
• Division Facts and Strategies Chapter Practice
• Division Facts and Strategies Cumulative Practice
• Division Facts and Strategies STEAM Performance Task

Lesson 4.1 Use Arrays to Divide

Explore and Grow

Build an array to model 12. Draw the array.
Number of rows: _______
Number in each row: ______
Answer:
Number of rows: 4
Number in each row: 3

Structure
Compare your array with your partner’s array. How are they the same? How are they different?
Answer: The number of counters is the same but the rows and columns are different.

Think and Grow: Division and Arrays

Example
There are 40 counters. The counters are in5 equal rows. How many counters are in each row?

There are ______ counters in each row.

Answer: 8

Explanation:
Number of rows = 5
Total number of counters = 40
So, the division equation is 40 ÷ 5 = 8
There are 8 counters in each row.

Show and Grow

Find the quotient.
Question 1.

14 ÷ 2 = _____
Answer: 7

Explanation:
Number of counters = 14
Number of rows = 2
To find the quotient you have to divide the number of counters by the number of rows.
14 ÷ 2 = 7
Thus the number of columns in each row = 7

Question 2.

12 ÷ 3 = ______
Answer: 4

Explanation:
Number of counters = 12
Number of rows = 3
To find the quotient you have to divide the number of counters by the number of rows.
12 ÷ 3 = 4
Thus the number of columns in each row = 4

Question 3.
There are 20 counters. The counters are in 4 equal rows. How many counters are in each row?
4 rows of _____
20 ÷ 4 = _____
Answer: 5

Explanation:
Given,
There are 20 counters. The counters are in 4 equal rows.
To find the quotient you have to divide the number of counters by the number of rows.
20 ÷ 4 = 5
4 rows of 5
There are 5 counters in each row.

Question 4.
You have 21 counters. You arrange them with 7 counters in each row. How many rows of counters do you make?
____ rows of 7
21 ÷ 7 = _____
Answer: 3

Explanation:
Given,
You have 21 counters. You arrange them with 7 counters in each row.
21 ÷ 7 = 3
3 rows 7
Thus you make 3 rows of 7 counters.

Apply and Grow: Practice

Question 5.
There are 25 counters. The counters are in 5 equal rows. How many counters are in each row?
5 rows of _____
25 ÷ 5 = _____
Answer: 5

Explanation:
Given,
There are 25 counters.
The counters are in 5 equal rows.
To find the quotient you have to divide the number of counters by the number of rows.
25 ÷ 5 = 5
There are 5 counters in each row.

Question 6.
There are 48 counters. The counters are in 8 equal rows. How many counters are in each row?
8 rows of _____
48 ÷ 8 = _____
Answer: 6

Explanation:
Given,
There are 48 counters. The counters are in 8 equal rows.
To find the quotient you have to divide the number of counters by the number of rows.
48 ÷ 8 = 6
8 rows of 6 counters
There are 6 counters in each row.

Question 7.
You have 42 counters. You arrange them with 6 counters in each row. How many rows of counters do you make?
____ rows of 6
42 ÷ 6 = _____
Answer: 7

Explanation:
Given,
You have 42 counters. You arrange them with 6 counters in each row.
7 rows of 6
42 ÷ 6 = 7
There are 7 rows of counters.

Question 8.
You have 27 counters. You arrange them with 9 counters in each row. How many rows of counters do you make?
____ rows of 9
27 ÷ 9 = _____
Answer: 3

Explanation:
Given that,
You have 27 counters. You arrange them with 9 counters in each row.
27 ÷ 9 = 3
3 rows of 9.
There are 3 rows of counters.

Write a division equation for the array.
Question 9.

_____ ÷ _____ = _____
Answer: 4 ÷ 2 = 2

Explanation:
There are 4 rows and 2 counters.
The division equation for the array is 4 ÷ 2 = 2

Question 10.

____ ÷ _____ = _____
Answer: 40 ÷ 4 = 10

Explanation:
There are 4 rows and 40 counters.
The division equation for the array is 40 ÷ 4 = 10

Question 11.
YOU BE THE TEACHER
Your friend has 63 counters in 7 equal rows. Your friend says that finding 7 ÷ 63 will give the number of columns. Is your friend correct? Explain.
Answer: No your friend is incorrect.

Explanation:
Given,
Your friend has 63 counters in 7 equal rows.
Your friend says that finding 7 ÷ 63 will give the number of columns.
Your friend is incorrect because you have to divide the number of counters by a number of equal rows.
63 ÷ 7 = 9
There are 9 columns.

Think and Grow: Modeling Real Life

Two groups of students are playing a flip and find game. Your group arranges 48 cards in 8 equal rows. The other group arranges 28 cards in 4 equal rows. Which group has rows with more cards?
Draw:
Division equations:
_______ has rows with more cards.

Explanation:
Given,
Two groups of students are playing a flip and find game.
Your group arranges 48 cards in 8 equal rows.
48 ÷ 8 = 6 in each row
The other group arranges 28 cards in 4 equal rows.
28 ÷ 4 = 7 in each row.
The second group has 7 rows with more cards.

Show and Grow

Question 12.
Newton arranges 12 magnets in 6 equal rows. Descartes arranges 15 magnets in 3 equal rows. Who has rows with more magnets?

Answer:
Given,
Newton arranges 12 magnets in 6 equal rows.
12 ÷ 6 = 2
Descartes arranges 15 magnets in 3 equal rows.
15 ÷ 3 = 5
Thus Descartes has 5 rows with more magnets.

Question 13.
You have 18 jars of green slime and 12 jars of purple slime at a party. Each guest takes 3 jars. There are none left. How many guests are at the party?

Answer:
Given that,
You have 18 jars of green slime and 12 jars of purple slime at a party.
18 + 12 = 30 jars
Each guest takes 3 jars. There are none left.
30 ÷ 3 = 10
Thus there are 10 guests at the party.

Use Arrays to Divide Homework & Practice 4.1

Find the quotient.
Question 1.

10 ÷ 2 = _____
Answer: 5

Explanation:
Number of counters = 10
Number of rows = 2
To find the quotient you have to divide the number of counters by the number of rows.
10 ÷ 2 = 5
Thus there are 5 counters in each row.

Question 2.

35 ÷ 5 = _____
Answer: 7

Explanation:
Number of counters = 35
Number of rows = 5
To find the quotient you have to divide the number of counters by the number of rows.
35 ÷ 5 = 7
Thus there are 7 counters in each row.

Question 3.
There are 9 counters. The counters are in 3 equal rows. How many counters are in each row?
3 rows of ____
9 ÷ 3 = ____
Answer: 3

Explanation:
Given that,
There are 9 counters. The counters are in 3 equal rows.
To find the quotient you have to divide the number of counters by the number of rows.
9 ÷ 3 = 3
3 rows of 3
Thus there are 3 counters in each row.

Question 4.
There are 60 counters. The counters are in 6 equal rows. How many counters are in each row?
6 rows of _____
60 ÷ 6 = ____
Answer: 10.

Explanation:
Given,
There are 60 counters.
The counters are in 6 equal rows.
To find the number of counters in each row you have to divide the total number of counters by the number of rows.
60 ÷ 6 = 10
6 rows of 10 counters
Thus there are 10 counters in each row.

Question 5.
You have 72 counters. You arrange them with 8 counters in each row. How many rows of counters do you make?
____ rows of 8
72 ÷ 8 = ____
Answer: 9

Explanation:
Given,
You have 72 counters. You arrange them with 8 counters in each row.
To find the number of counters in each row you have to divide the total number of counters by the number of rows.
72 ÷ 8 = 9
9 rows of 8.
Thus there are 9 counters in each row.

Question 6.
You have 24 counters. You arrange them with 6 counters in each row. How many rows of counters do you make?
____ rows of 6
24 ÷ 6 = ____
Answer: 4

Explanation:
Given,
You have 24 counters. You arrange them with 6 counters in each row.
24 ÷ 6 = 4
4 rows of 6
Thus there are 4 counters in each row.

Question 7.
Writing
How can you use an array to find 35 ÷ 7?
Answer: 5

Explanation:
Number of counters = 35
Number of rows = 7
To find the number of counters in each row you have to divide the total number of counters by the number of rows.
35 ÷ 7 = 5

Question 8.
Precision
Label the parts of the division problem using quotient, dividend and divisor.

Answer:
12 is dividend
6 is divisor and
2 is quotient

Question 9.
Modeling Real Life
Your apartment building has 40 mailboxes in 5 equal rows. Your friend’s apartment building has 40 mailboxes in 4 equal rows. Which apartment building has rows with more mailboxes?
Answer: Your friend’s apartment building 10 rows with more mailboxes

Explanation:
Given,
Your apartment building has 40 mailboxes in 5 equal rows.
40 ÷ 5 = 8
Your friend’s apartment building has 40 mailboxes in 4 equal rows.
40 ÷ 4 = 10
Thus Your friend’s apartment building 10 rows with more mailboxes.

Question 10.
Modeling Real Life
There are 8 snack bags of pretzels and 10 snack bags of popcorn. The bags are divided equally between 9 friends. How many snack bags does each friend get?

Answer:
Given,
There are 8 snack bags of pretzels and 10 snack bags of popcorn.
8 + 10 = 18 snack bags
The bags are divided equally between 9 friends.
18 ÷ 9 = 2
Thus each friend gets 2 snack bags.

Review & Refresh

Find the missing factor
Question 11.
____ × 2 = 20
Answer: 10

Explanation:
Let the missing factor be a.
a × 2 = 20
a = 20/2
a = 10
Thus the missing factor is 10.

Question 12.
1 × ____ = 5
Answer: 5

Explanation:
Let the missing factor be b.
1 × b = 5
b = 5/1
b = 5
Thus the missing factor is 5.

Question 13.
____ × 6 = 0
Answer: 0

Explanation:
Let the missing factor be c.
c × 6 = 0
c = 0/6
c = 0
Thus the missing factor is 0.

Question 14.
1 × ___ = 0
Answer: 0

Explanation:
Let the missing factor be d.
1 × d = 0
d = 0/1
d = 0
Thus the missing factor is 0.

Question 15.
____ × 4 = 40
Answer: 10

Explanation:
Let the missing factor be e.
e × 4 = 40
e = 40/4
e = 10
Thus the missing factor is 10.

Question 16.
7 × ____ = 7
Answer: 1

Explanation:
Let the missing factor be f.
7 × f = 7
f = 7/7
f = 1
Thus the missing factor is 1.

Lesson 4.2 Relate Multiplication and Division

Explore and Grow

Use 24 counters to make an array. Draw the array. Write a multiplication equation and a division equation for the array.
____ × ____ = _____
_____ ÷ ____ = ____
Answer: 6 × 4 = 24
24 ÷ 6 = 4

Explanation:

The division equation is 24 ÷ 6 = 4 or 24 ÷ 4 = 6

Structure
Compare your equations to your partner’s equations. How are they the same? How are they different?
Answer:
Both the equations are the same but the rows and columns are different.

Think and Grow: Multiplication and Division

A fact family is a group of related facts that uses the same numbers.

Answer:
Multiplication:
5 rows of 6 counters
5 × 6 = 30
30 counters
Division:
30 counters in 5 equal rows
30 ÷ 5 = 6
60 counters in each row.
Fact family for 5, 6 and 30:
5 × 6 = 30
6 × 5 = 30
30 ÷ 5 = 6
30 ÷ 6 = 5

Show and Grow

Use the array to complete the equations.
Question 1.

3 × ____ = 24
24 ÷ 3 = ____
Answer: 8

Explanation:
Number of rows = 3
Number of counters = 24
To find the number of counters in each row you have to divide the total number of counters by the number of rows.
24 ÷ 3 = 8
3 × x = 24
x = 24/3
x = 8

Question 2.

4 × ____ = 20
20 ÷ 4 = _____
Answer: 5

Explanation:
Number of rows = 4
Number of counters = 20
To find the number of counters in each row you have to divide the total number of counters by the number of rows.
4 × x = 20
x = 20/4
x = 5
20 ÷ 4 = 5

Question 3.
Draw an array to find 2 × 7. Write the other facts in the fact family.
2 × 7 = _____
___________
___________
___________
Answer: 14

Explanation:

The fact family for the numbers 2, 7 and 14
2 × 7 = 14
7 × 2 = 14
14 ÷ 2 = 7
14 ÷ 7 = 2

Apply and Grow: Practice

Question 4.
Draw an array to find 3 × 6.
Write the other 3 facts in the fact family.
3 × 6 = ______
____________
_____________
____________
Answer: 18

Explanation:

The fact family for 3, 6 and 18 are
3 × 6 = 18
6 × 3 = 18
18 ÷ 3 = 6
18 ÷ 6 = 3

Complete the fact family.
Question 5.
7 × ____ = 70
____ × 7 = 70
70 ÷ 10 = ____
70 ÷ ___ = 10
Answer:
The fact family for 7, 10 and 70 are:
7 × 10 = 70
10 × 7 = 70
70 ÷ 10 = 7
70 ÷ 7 = 10

Question 6.
5 × ____ = 40
____ × 5 = 40
40 ÷ 8 = _____
40 ÷ ____ = 8
Answer:
The fact family for 5, 8 and 40 are:
5 × 8 = 40
8 × 5 = 40
40 ÷ 8 = 5
40 ÷ 5 = 8

Write the fact family for the numbers.
Question 7.
2, 5, 10
Answer:
The fact family for 2, 5 and 10 are:
2 × 5 = 10
5 × 2 = 10
10 ÷ 2 = 5
10 ÷ 5 = 2

Question 8.
4, 3, 12
Answer:
The fact family for 3, 4 and 12 are:
4 × 3 = 12
3 × 4 = 12
12 ÷ 3 = 4
12 ÷ 4 = 3

Question 9.
Structure
Find each product. Then match the multiplication fact with the related division fact.

Answer:

Question 10.
DIG DEEPER!
Is 4 × 6 = 24 part of the fact family for 3 × 8 = 24? Explain.
Answer: No. Because the rows and columns are different.

Think and Grow: Modeling Real Life

Your teacher divides the items shown equally among 9 students. Write two equations that you can use to show how many straws each student gets.

Division equation:
Multiplication equation:

Answer:
Your teacher divides the items shown equally among 9 students.
Straws – 54
54 ÷ 9 = 6
Thus each student gets 6 straws.
Division Equation is 54 ÷ 9 = 6
Multiplication equation 9 × 6 = 54

Show and Grow

Question 11.
Use the table above to write two equations that you can use to show how many containers of clay each student gets.
Answer:
Your teacher divides the items shown equally among 9 students.
Containers of the day – 27
27 ÷ 9 = 3
Division Equation is 27 ÷ 9 = 3
Multiplication equation is 3 × 9 = 27

Question 12.
Use the table above to find how many more toothpicks students will get than straws.

Answer:
Your teacher divides the items shown equally among 9 students.
Toothpicks – 72
72 ÷ 9 = 8
Each student gets 8 toothpicks
Straws – 54
54 ÷ 9 = 6
Thus each student gets 6 straws.
8 – 6 = 2
Thus Each student will get 2 toothpicks more than the straws.

Question 13.
Explain how a multiplication fact can help you solve 30 ÷ 3 = _____.
Answer: 10 × 3 = 30

Explanation:
The multiplication fact to help you to solve 30 ÷ 3 is 10 × 3.
By using the multiplication fact you can get the solution for 30 ÷ 3 = 10

Relate Multiplication and Division Homework & Practice 4.2

Use the array to complete the equations
Question 1.

2 × ___ = 8
8 ÷ 2 = _____
Answer: 4

Explanation:
Number of counters = 8
Number of rows = 2
8 ÷ 2 = 4
2 × x = 8
x = 8/2
x = 4

Question 2.

3 × ____ = 9
9 ÷ 3 = ____
Answer: 3

Explanation:
Number of counters = 9
Number of rows  = 3
3 × x = 9
x = 9/3
x = 3
9 ÷ 3 = 3

Question 3.
Draw an array to find 5 × 7. Write the other 3 facts in the fact family.
5 × 7 = _____
___________
___________
___________
Answer: 35

Explanation:
The other 3 facts are
5 × 7 = 35
7 × 5 = 35
35 ÷ 5 = 7
35 ÷ 7 = 5

Complete the fact family
Question 4.

Answer: 1, 9

Explanation:
9 × 1 = 9
9 ÷ 1 = 9
1 × 9 = 9
9 ÷ 9 = 1

Question 5.

Answer: 7, 6

Explanation:
6 × 7 = 42
42 ÷ 7 = 6
7 × 6 = 42
42 ÷ 6 = 7

Write the fact family for the numbers.
Question 6.
4, 8, 32
Answer: 8 × 4 = 32, 32 ÷ 4 = 8, 32 ÷ 8 = 4

Explanation:
The facts family for the numbers are
4 × 8 = 32
8 × 4 = 32
32 ÷ 8 = 4
32 ÷ 4 = 8

Question 7.
6, 7, 42
Answer: 6 × 7 = 42, 42 ÷ 6 = 7, 42 ÷ 7 = 6

Explanation:
The facts family for the numbers are
6 × 7 = 42
7 × 6 = 42
42 ÷ 7 = 6
42 ÷ 6 = 7

Question 8.
Which One Doesn’t Belong?
Which equation does not belong with the other three?
3 × 7 = 21
7 × 3 = 21
21 ÷ 7 = 3
7 + 3 = 10
Answer: 7 + 3 = 10

Explanation:
The equation that does not belong to other three equation is 7 + 3 = 10.

Question 9.
DIG DEEPER!
Newton has 16 pennies. He wants to put them in stacks that are the same height. How many stacks does he make?

Answer: 2

Explanation:
Given,
Newton has 16 pennies. He wants to put them in stacks that are the same height.
16 ÷ 8 = 2
Thus he makes 2 stacks.

Question 10.
Modeling Real Life
Your art teacher divides the items shown equally among 6 students. Write two equations that you can use to show how many pieces of paper each student gets.

Answer:
Given
Your art teacher divides the items shown equally among 6 students.
48 ÷ 6 = 8 paintbrushes
42 ÷ 6 = 7 pieces of paper
Thus each student gets 7 pieces of paper.

Question 11.
Modeling Real Life
Use the table above to find how many more paintbrushes students will get than paint trays.
Answer:
Your art teacher divides the items shown equally among 6 students.
18 ÷ 6 = 3 paint trays
48 ÷ 6 = 8 paintbrushes
8 – 3 = 5
Thus the students will get 5 more paintbrushes than paint trays.

Review & Refresh

Find the missing factor.
Question 12.
5 × ____ = 45
Answer: 9

Explanation:

Let the missing factor be g.
5 × g = 45
g = 45/5
g = 9
Thus the missing factor is 9.

Question 13.
2 × ____ = 16
Answer: 8

Explanation:
Let the missing factor be h.
2 × h = 16
h = 16/2
h = 8
Thus the missing factor is 8.

Question 14.
_____ × 2 = 4
Answer: 2

Explanation:
Let the missing factor be i.
i × 2 = 4
i = 4/2
i = 2
Thus the missing factor is 2.

Question 15.
____ × 3 = 15
Answer: 5

Explanation:
Let the missing factor be j.
j × 3 = 15
j = 15/3
j = 5
Thus the missing factor is 5.

Question 16.
5 × ____ = 10
Answer: 2

Explanation:
Let the missing factor be k.
5 × k = 10
k = 10/5
k = 2
Thus the missing factor is 2.

Question 17.
5 × ____ = 5
Answer: 1

Explanation:
Let the missing factor be l.
5 × l = 5
l = 5/5
l = 1
Thus the missing factor is 1.

Lesson 4.3 Divide by 2, 5, or 10

Explore and Grow

Use the number line to model 10 ÷ 2.

10 ÷ 2 = _____
Answer: 5

Structure
In your number line model, does 2 represent the number of equal groups or the size of the groups? Explain.
Answer: Yes, the number line model 2 represents the number of equal groups or the size of the groups.

Think and Grow: Divide by 2, 5, or 10

Example
Find 16 ÷ 2.

Answer:
Number of rows = 2
Number of counters = 16
Division equation is 16 ÷ 2 = 8
Multiplication equation is 2 × 8 = 16

Example
Find 20 ÷ 5.
Think: 5 times what number is 20?
5 × ____ = 20

Answer:
Number of rows = 5
Number of counters = 20
Division equation is 20 ÷ 5 = 4
Multiplication equation is 5 × 4 = 20

Example
Find 30 ÷ 10.
Think: 10 times what number is 30?
10 × _____ = 30

Answer:
Number of rows = 10
Number of counters = 30
Division equation is 30 ÷ 10 = 3
Multiplication equation is 10 × 3 = 30

Show and Grow

Write the related multiplication fact. Then find the quotient
Question 1.
Find 60 ÷ 10.
10 × ____ = 60
60 ÷ 10 = ____
Answer: 6, 6

Explanation:
The related multiplication fact for 60 and 10 is
10 × x = 60
x = 60/10
x = 6
60 ÷ 10 = 6
Thus the quotient is 6.

Question 2.
Find 14 ÷ 2
2 × ____ = 14
14 ÷ 2 = _____
Answer: 7, 7

Explanation:
The related multiplication fact for 14 and 2 is
2 × x = 14
x = 14/2
x = 7
14 ÷ 2 = 7
Thus the quotient is 7.

Question 3.
Find 35 ÷ 5
5 × ___ = 35
35 ÷ 5 = ____
Answer: 7, 7

Explanation:
The related multiplication fact for 5 and 35 is
5 × x = 35
x = 35/5
x = 7
35 ÷ 5 = 7
Thus the quotient is 7.

Write the related multiplication fact. Then find the quotient
Question 4.
Find 4 ÷ 2.
2 × ____ = 4
4 ÷ 2 = ____
Answer: 2, 2

Explanation:
The related multiplication fact for 2 and 4 is
2 × x = 4
x = 4/2
x = 2
4 ÷ 2 = 2
Thus the quotient is 2.

Question 5.
Find 15 ÷ 5.
5 × ____ = 15
15 ÷ 5 = ____
Answer: 3, 3

Explanation:
The related multiplication fact for 5 and 15 is
5 × x = 15
x = 15/5
x = 3
15 ÷ 5 = 3
Thus the quotient is 3.

Question 6.
Find 10 ÷ 10.
10 × ____ = 10
10 ÷ 10 = ____
Answer: 1, 1

Explanation:
The related multiplication fact for 10 and 10 is
10 × x = 10
x = 10/10
x = 1
10 ÷ 10 = 1
Thus the quotient is 1.

Find the quotient
Question 7.
70 ÷ 10 = _____
Answer: 7
Divide both the numbers 70 and 10
70/10 = 7
Thus 7 is the quotient.

Question 8.
25 ÷ 5 = _____
Answer: 5
Divide both the numbers 25 and 5.
25/5 = 5
Thus 5 is the quotient.

Question 9.
18 ÷ 2 = ____
Answer: 6
Divide both the numbers 18 and 2.
18/2 = 9
Thus 9 is the quotient.

Question 10.

Answer: 6
Divide both the numbers 30 and 5.
30/5 = 6
Thus 6 is the quotient.

Question 11.

Answer: 8
Divide both the numbers 16 and 2.
16/2 = 8
Thus 8 is the quotient.

Question 12.

Answer: 9
Divide both the numbers 10 and 90.
90/10 = 9
Thus 9 is the quotient.

Question 13.
Divide 60 by 10.
Answer: 6
Divide both the numbers 60 and 10.
60/10 = 6
Thus 6 is the quotient.

Question 14.
Divide 14 by 2.
Answer: 7
Divide both the numbers 2 and 14.
14/2 = 7
Thus 7 is the quotient.

Question 15.
Divide 45 by 5.
Answer: 9
Divide both the numbers 45 and 5.
45/5 = 9
Thus 9 is the quotient.

Find the missing divisor.
Question 16.
12 ÷ ____ = 6
Answer: 2

Explanation:
Let the missing divisor is x
12 ÷ x = 6
12/x = 6
12 = 6 × x
x = 12/6
x = 2
Thus the missing divisor is 2.

Question 17.
10 ÷ ____ = 2
Answer: 5

Explanation:
Let the missing divisor is y
10 ÷ y = 2
10/y = 2
10 = 2 × y
y = 10/2
y = 5
Thus the missing divisor is 5.

Question 18.
50 ÷ ____ = 5
Answer: 10

Explanation:
Let the missing divisor is z.
50 ÷ z = 5
50/z = 5
50 = 5 × z
z = 50/5
z = 10
Thus the missing divisor is 10.

Question 19.
You make 2 batches of pancakes. You use 6 cups of flour. How many cups of flour are in 1 batch?

Answer: 3

Explanation:
Given that,
You make 2 batches of pancakes.
You use 6 cups of flour.
6/2 = 3 cups of flour
There are 3 cups of flour in 1 batch.

Question 20.
DIG DEEPER!
I am an even number. If you multiply me by 5, then divide the product by 10, the quotient is 2. What number am I?
Answer: 20

Explanation:
I am an even number. If you multiply me by 5, then divide the product by 10, the quotient is 2.
Let 4 be the even number.
4 × 5 = 20
20/10 = 2
Thus the number is 4.

Think and Grow: Modeling Real Life

Fourteen students say a dog is their favorite pet. How many symbols should you draw to complete the picture graph?

Division equation:
You should draw _____ symbols

Answer:
Given,
Fourteen students say a dog is their favorite pet.

???? = 2 students
14 ÷ 2 = 7 pictures

Show and Grow

Question 21.
Twenty-five students say riding a bike is their favorite summer activity. How many symbols should you draw to complete the picture graph?

Answer:
Given,
Twenty-five students say riding a bike is their favorite summer activity.
???? = 5 students
25 = ???? ????????????????
25 ÷ 5 = 5

Question 22.
Erasers cost 10¢ each. How many erasers can you buy with 70¢?
Answer:
Given that,
Erasers cost 10¢ each.
You need to buy with 70¢
10 × x = 70
x = 70/10
x = 7
Therefore you can buy 7 erasers for 70¢.

Question 23.
You have 26 red linking cubes and 24 blue linking cubes. You use all of the linking cubes to make towers with 10 linking cubes each. How many towers do you make?
Answer: 5

Explanation:
Given that,
You have 26 red linking cubes and 24 blue linking cubes.
26 + 24 = 50 linking cubes
You use all of the linking cubes to make towers with 10 linking cubes each.
50 ÷ 10 = 5 towers
Thus you can make 5 towers.

Divide by 2, 5, or 10 Homework & Practice 4.3

Write the related multiplication fact. Then find the quotient.
Question 1.
Find 20 ÷ 10
10 × ____ = 20
20 ÷ 10 = ____
Answer: 2, 2

Explanation:
Let x be the unknown factor.
10 × x  = 20
x = 20/10
x = 2
First, we need to check whether the divisor or dividend is the related multiplication fact or not.
Next check whether the divisor or the dividend the product in the related multiplication fact or not.
If both are the same then the quotient is the unknown factor.
20/10 = 2
Thus the quotient is 2.

Question 2.
Find 8 ÷ 2.
2 × ___ = 8
8 ÷ 2 = ____
Answer: 4, 4

Explanation:
Let y be the unknown factor.
First, we need to check whether the divisor or dividend is the related multiplication fact or not.
Next check whether the divisor or the dividend the product in the related multiplication fact or not.
If both are the same then the quotient is the unknown factor.
2 × y = 8
y = 8/2
y = 4
Thus the quotient is 4.

Question 3.
Find 50 ÷ 5
5 × ___ = 50
50 ÷ 5 = ____
Answer: 10, 10

Explanation:
Let z be the unknown factor.
First, we need to check whether the divisor or dividend is the related multiplication fact or not.
Next check whether the divisor or the dividend the product in the related multiplication fact or not.
If both are the same then the quotient is the unknown factor.
5 × z = 50
z = 50/5
z = 10
Thus the quotient is 10.

Find the quotient
Question 4.
100 ÷ 10 = _____
Answer: 10

Explanation:
100/10 = 10
The quotient is 10.

Question 5.
45 ÷ 5 = _____
Answer: 9

Explanation:
45/5 = 9
The quotient is 9.

Question 6.
14 ÷ 2 = _____
Answer: 7

Explanation:
14/2 = 7
The quotient is 7.

Question 7.

Answer: 6

Explanation:
12/2 = 6
The quotient is 6.

Question 8.

Answer: 6

Explanation:
60/10 = 6
The quotient is 6.

Question 9.

Answer: 2

Explanation:
10/5 = 2
The quotient is 2.

Find the missing divisor
Question 10.
20 ÷ ___ = 5
Answer: 4

Explanation:
Let x be the missing divisor
20 ÷ x = 5
20/x = 5
20 = x × 5
x = 20/5
x = 4
Thus the missing divisor is 4.

Question 11.
40 ÷ ____ = 10
Answer: 4

Explanation:
Let x be the missing divisor.
40 ÷ x = 10
40/x = 10
40 = 10 × x
x = 40/10
x = 4
Thus the missing divisor is 4.

Question 12.
4 ÷ ____ = 2
Answer: 2

Explanation:
Let x be the missing divisor.
4 ÷ x = 2
4/x = 2
4 = 2 × x
x = 4/2
x = 2
Thus the missing divisor is 2.

Question 13.
Number Sense
The American flag has 50 stars. It has 10 times as many stars as the Chinese flag. How many stars are on the Chinese flag?
Answer:
Given,
The American flag has 50 stars. It has 10 times as many stars as the Chinese flag.
50/10 = 5
Thus there are 5 stars on the Chinese flag.

Question 14.
Open-Ended Write a division equation for each description.

Answer: 10 ÷ 2 = 5
In this equation 2 is the divisor, 5 is the quotient and 10 is the dividend.

Question 15.
Repeated Reasoning
Complete the table.

Answer:

Question 16.
You buy 20 flowers. You want an equal number of flowers in each of the 5 pots. How many flowers do you put in each pot?

Answer:
Given,
You buy 20 flowers. You want an equal number of flowers in each of the 5 pots.
20/5 = 4
Thus there are 4 flowers in each pot.

Question 17.
Modeling Real Life
Fifty students say they have traveled on an airplane. How many symbols should you draw to complete the picture graph?

Answer:
Given that,
Fifty students say they have traveled on an airplane.
???? = 10 students
50 = 10 × 5
= ???? × 5

Question 18.
Modeling Real Life
A jeweler has 17 gold rings and 18 silver rings. She puts them in a ring tray in 5 rows. How many rings are in each row?
Answer:  7 rings

Explanation:
Given that,
A jeweler has 17 gold rings and 18 silver rings.
17 + 18 = 35
She puts them in a ring tray in 5 rows.
35/5 = 7
Therefore there are 7 rings in each row.

Review & Refresh

Compare.
Question 19.

Answer: >

Explanation:
8 × 3 = 24
7 × 3 = 21
24 > 21
So, 8 × 3 > 7 × 3

Question 20.

Answer: =

Explanation:
5 × 3 = 15
15 = 15
So, 15 = 5 × 3

Question 21.

Answer: <

Explanation:
3 × 3 = 9
4 × 3 = 12
9 < 12
So, 3 × 3 < 4 × 3

Lesson 4.4 Divide by 3 or 4

Explore and Grow

Put 12 counters into 3 equal groups. Draw to show your groups.
Use your equal groups to help you find the quotient.
12 ÷ 3 = ____
Answer: 4

Structure
Put 12 counters into 4 equal groups. Draw to show your groups. Write a division equation to match. What do you notice?
Answer: 3

Think and Grow: Divide by 3 or 4

Example
Find 18 ÷ 3

Answer: 6

Explanation:
Let the unknown number be x.
18 ÷ 3 = x
18/3 = x
x = 6
Thus 18 is 3 times 6.

Example
Find 32 ÷ 4.

Answer: 8

Explanation:
Let the unknown number be x.
32 ÷ 4 = x
32/4 = x
x = 8
Thus 32 is 4 times 8.
4 × 8 = 32

Show and Grow

Complete the model and find the quotient.
Question 1.
Find 28 ÷ 4.

28 ÷ 4 = _____
Answer: 7

Explanation:
Number of rows = 4
Number of counters = 28
Divide the number of counters by the number of rows
28/4 = 7
Thus there are 7 columns in each row.

Question 2.
Find 9 ÷ 3.

9 ÷ 3 = _____
Answer: 3

Explanation:

Number of rows = 3
Number of counters = 9
Divide the number of counters by the number of rows
9/3 = 3
Thus there are 3 columns in each row.

Apply and Grow: Practice

Complete the model and find the quotient.
Question 3.
Find 8 ÷ 4.

8 ÷ 4 = _____
Answer: 2

Explanation:

Number of rows = 4
Number of counters = 8
Divide the number of counters by the number of rows
8/4 = 2
Thus there are 2 columns in each row.

Question 4.
Find 24 ÷ 3.

24 ÷ 3 = _____
Answer: 8

Explanation:

Number of rows = 3
Number of counters = 24
Divide the number of counters by the number of rows
24/3 = 8
Thus there are 8 columns in each row.

Find the quotient
Question 5.
12 ÷ 3 = _____
Answer: 4

Explanation:
Divide the two numbers 12 and 3.
12/3 =4
Thus the quotient is 4

Question 6.
20 ÷ 4 = _____
Answer: 5

Explanation:
Divide the two numbers 20 and 4.
20/4 = 5
Thus the quotient is 5.

Question 7.
15 ÷ 3 = _____
Answer: 5

Explanation:
Divide the two numbers 15 and 3
15/3 = 5
Thus the quotient is 5.

Question 8.

Answer: 8

Explanation:
Divide the two numbers 32 and 4
32/4 = 8
Thus the quotient is 8.

Question 9.

Answer: 6

Explanation:
Divide the two numbers 3 and 18
18/3 = 6
Thus the quotient is 6.

Question 10.

Answer: 9

Explanation:
Divide the two numbers 36 and 4
36/4 = 9
Thus the quotient is 9.

Question 11.
Divide 9 by 3.
Answer: 3

Explanation:
Divide the two numbers 3 and 9
9/3 = 3
Thus the quotient is 3.

Question 12.
Divide 12 by 4.
Answer: 3

Explanation:
Divide the two numbers 12 and 4.
12/4 = 3
Thus the quotient is 3.

Question 13.
Divide 21 by 3.
Answer: 7

Explanation:
Divide the two numbers 21 by 3
21/3 = 7
Thus the quotient is 7.

Compare
Question 14.

Answer: >

Explanation:
Divide the two numbers.
27 ÷ 3 = 9
28 ÷ 4 = 7
9 > 7
27 ÷ 3 > 28 ÷ 4

Question 15.

Answer: >

Explanation:
Divide the two numbers
30 ÷ 3 =10
24 ÷ 4 = 6
10 > 6
30 ÷ 3 > 24 ÷ 4

Question 16.

Answer: <

Explanation:
Divide the two numbers
6 ÷ 3 = 2
16 ÷ 4 = 4
2 < 4
6 ÷ 3 < 16 ÷ 4

Question 17.
There are 36 water bottles in a package. The bottles are in 4 rows. How many water bottles are in each row?
Answer: 9

Explanation:
Given,
There are 36 water bottles in a package.
The bottles are in 4 rows.
36/4 = 9
Thus 9 water bottles are in each row.

Question 18.
DIG DEEPER!
Can you divide 20 students into 3 equal groups? Explain.
Answer: No
20 is not the multiple of 3. So, 20 students cannot be divided into 3 equal groups.

Think and Grow: Modeling Real Life

You arrange 36 chairs in 4 equal rows. You arrange 21 music stands in 3 equal rows. How many more chairs are in each row than music stands?

Division equations:
There are ______ more chairs in each row.

Answer:
Given,
You arrange 36 chairs in 4 equal rows.
36/4 = 9 chairs in each row
You arrange 21 music stands in 3 equal rows.
21/3 = 7 chairs in each row
Thus there are 2 more chairs in each row.

Show and Grow

Question 19.
You arrange 30 cups of fruit punch in 3 equal rows. You arrange 24 cups of lemonade in 4 equal rows. How many more cups of fruit punch are in each row than cups of lemonade?
Answer:
Given,
You arrange 30 cups of fruit punch in 3 equal rows.
30/3 = 10 cups of fruit punch
You arrange 24 cups of lemonade in 4 equal rows.
24/4 = 6 cups of lemonade
10 > 6
Thus 4 more cups of fruit punch are in each row than cups of lemonade.

Question 20.
You have a bag of 25 carrot sticks. You eat 5 of them and equally share the rest with 4 friends. How many carrot sticks does each friend get?

Answer: 5

Explanation:
Given that,
You have a bag of 25 carrot sticks. You eat 5 of them and equally share the rest with 4 friends.
25 – 5 = 20 carrot sticks
20/4 = 5 carrot sticks
Thus each friend gets 5 carrot sticks.

Question 21.
Your teacher has 18 yellow pencils and 18 red pencils. She puts 3 pencils on each desk in the class. How many desks are in the class?
Answer: 12

Explanation:
Given that,
Your teacher has 18 yellow pencils and 18 red pencils.
18 + 18 = 36 pencils
She puts 3 pencils on each desk in the class.
36/3 = 12
Thus there are 12 desks in the class.

Divide by 3 or 4 Homework & Practice 4.4

Complete the model and find the quotient
Question 1.
Find 27 ÷ 3

27 ÷ 3 = ______
Answer: 9

Explanation:
Number of rows = 3
Number of counters = 27
Divide the number of counters by the number of rows
27/3 = 9
Thus there are 9 counters in each row.

Question 2.
Find 20 ÷ 4.

20 ÷ 4 _____
Answer: 5

Explanation:

Number of rows = 4
Number of counters = 20
Divide the number of counters by the number of rows
20/4 = 5
Thus there are 5 counters in each row.

Find the quotient
Question 3.
6 ÷ 3 = _____
Answer: 2

Explanation:
Divide the two numbers 3 and 6.
6/3 = 2
Thus the quotient is 2.

Question 4.
28 ÷ 4 = _____
Answer: 7

Explanation:
Divide the two numbers 28 and 4.
28/4 = 7
Thus the quotient is 7.

Question 5.
18 ÷ 3 = _____
Answer: 6

Explanation:
Divide the two numbers 18 and 3.
18/3 = 6
Thus the quotient is 6.

Question 6.

Answer: 2

Explanation:
Divide the two numbers 4 and 8.
8/4 = 2
Thus the quotient is 2.

Question 7.

Answer: 8

Explanation:
Divide the two numbers 24 and 3
24/3 = 8
Thus the quotient is 8.

Question 8.

Answer: 4

Explanation:
Divide the two numbers 16 and 4.
16/4 = 4
Thus the quotient is 4.

Compare
Question 9.

Answer: <

Explanation:
Divide the two numbers.
9 ÷ 3 = 3
15 ÷ 3 = 5
3 < 5
So, 9 ÷ 3 < 15 ÷ 3

Question 10.

Answer: <

Explanation:
Divide the two numbers.
12 ÷ 3 = 4
12 ÷ 4 = 3
4 > 3
12 ÷ 3 > 12 ÷ 4

Question 11.

Answer: <

Explanation:
Divide the two numbers.
32 ÷ 4 = 8
36 ÷ 4 = 9
8 < 9
32 ÷ 4 < 36 ÷ 4

Question 12.
Number Sense
Tissue boxes are sold in packs of 3. A doctor’s office needs 21 boxes. How many packs should the office buy?
Answer: 7

Explanation:
Given that,
Tissue boxes are sold in packs of 3.
A doctor’s office needs 21 boxes.
21/3 = 7 packs
Therefore the office should buy 7 packs.

Question 13.
Structure
Write the division equation represented by the number line.

Answer: 32 ÷ 4 = 8

Explanation:
The count starts from 0.
The count jumps for every 4s.
There are 8 jumps.
Thus the division equation is 32 ÷ 4 = 8

Question 14.
YOU BE THE TEACHER
Your friend says she can use 3 × 6 = 18 to help find 6 ÷ 3. Is your friend correct? Explain.
Answer: Your friend is incorrect

Explanation:
Your friend says she can use 3 × 6 = 18 to help find 6 ÷ 3.
6 ÷ 3 = 2
The multiplication equation is different from the division equation.
By this, you can say that your friend is incorrect.

Question 15.
Modeling Real Life
A food vending machine has 40 snacks in rows of 4. A drink vending machine has 21 drinks in rows of 3. How many more rows of snacks are there than rows of drinks?
Answer:
Given,
A food vending machine has 40 snacks in rows of 4.
40/4 = 10
A drink vending machine has 21 drinks in rows of 3.
21/3 = 7
10 – 7 = 3
Thus 3 more rows of snacks are there than rows of drinks.

Question 16.
Modeling Real Life
A delivery person has 13 large packages and 14 small packages. He delivers 3 packages to each house. There are none left. How many houses does he deliver to?

Answer: 9

Explanation:
Given,
A delivery person has 13 large packages and 14 small packages.
13 + 14 = 27
He delivers 3 packages to each house. There are none left.
27/3 = 9
Thus he delivers 9 houses.

Review & Refresh

Find the product
Question 17.

Answer: 48

Explanation:
Multiply the two numbers 8 and 6.
8 × 6 = 48

Question 18.

Answer: 0

Explanation:
Multiply the two numbers 6 and 0.
Any number multiplied by 0 will be always 0.
6 × 0 = 0

Question 19.

Answer: 12

Explanation:
Multiply the two numbers 6 and 2.
6 × 2 = 12

Question 20.

Answer: 60

Explanation:
Multiply the two numbers 10 and 6.
10 × 6 = 60

Question 21.

Answer: 42

Explanation:
Multiply the two numbers 7 and 6.
7 × 6 = 42

Lesson 4.5 Divide by 6 or 7

Explore and Grow

Complete the statements and the models.

Answer: 42 ÷ 6 = 7

Reasoning
Without solving, which quotient is greater? Explain how you know.
Answer: The second figure is having a greater quotient.
42 ÷ 6 = 7
42 ÷ 7 = 6

Think and Grow: Divide by 6 or 7

Example
Find 48 ÷ 6.
Think: 6 times what number is 48?
6 × _____ = 48

Answer: 8

Explanation:
Number of rows = 6
Number of counters = 48
Divide the number of counters by the number of rows.
48/6 = 8
There are 8 columns.
6 × 8 = 48

Example
Find 49 ÷ 7.
Think: 7 times what number is 49?
7 × ____ = 49

Answer: 7

Explanation:
Number of rows = 7
Number of counters = 49
Divide the number of counters by the number of rows
49/7 = 7
There are 7 columns.
7 × 7 = 49

Show and Grow

Complete the model and find the quotient
Question 1.
Find 28 ÷ 7.

28 ÷ 7 = _____
Answer: 4

Explanation:
Number of rows = 7
Number of counters = 28
Divide the number of counters by the number of rows
28/7 = 4
There are 4 columns.

Question 2.
Find 54 ÷ 6.

54 ÷ 6 = _____
Answer: 9

Explanation:
Number of rows = 6
Number of counters = 54
Divide the number of counters by the number of rows
54/6 = 9
There are 9 columns.

Apply and Grow: Practice

Complete the model and find the quotient
Question 3.
Find 36 ÷ 6.

36 ÷ 6 = _____
Answer: 6

Explanation:
Number of rows = 6
Number of counters = 36
Divide the number of counters by the number of rows
36/6 = 6
There are 6 columns.

Question 4.
Find 14 ÷ 7.

14 ÷ 7 = _____
Answer: 2

Explanation:
Number of rows = 7
Number of counters = 14
Divide the number of counters by the number of rows
14/7 = 2
There are 2 columns.

Find the quotient
Question 5.
60 ÷ 6 = _____
Answer: 10

Explanation:
Divide the two numbers 60 and 6.
60/6 = 10
Thus the quotient is 10.

Question 6.
35 ÷ 7 = _____
Answer: 5

Exp7lanation:
Divide the two numbers 35 and 7.
35/7 = 5
Thus the quotient is 5.

Question 7.
24 ÷ 6 = _____
Answer: 4

Explanation:
Divide the two numbers 24 and 6.
24/6 = 4
Thus the quotient is 4.

Question 8.

Answer: 2

Explanation:
Divide the two numbers 12 and 6.
12/6 = 2
Thus the quotient is 2.

Question 9.

Answer: 6

Explanation:
Divide the two numbers 42 and 7.
42/7 = 6
Thus the quotient is 6.

Question 10.

Answer: 9

Explanation:
Divide the two numbers 63 and 7
63/7 = 9
Thus the quotient is 9.

Find the missing divisor
Question 11.
28 ÷ ____ = 7
Answer: 4

Explanation:
Let the missing divisor be a.
28 ÷ a = 7
28/a = 7
28 = 7a
a = 28/7
a = 4
Thus the missing divisor is 4.

Question 12.
30 ÷ ____ = 6
Answer: 5

Explanation:
Let the missing divisor be b.
30 ÷ b = 6
30/b = 6
30 = 6 × b
b = 30/6
b = 5
Thus the missing divisor is 5.

Question 13.
70 ÷ _____ = 7
Answer: 10

Explanation:
Let the missing divisor be c.
70 ÷ c = 7
70/c = 7
70 = 7 × c
c = 70/7
c = 10
Thus the missing divisor is 10.

Question 14.
You have 24 stones for the game mancala. There are 6 holes on the board. Each hole gets an equal number of stones. How many stones do you put in each hole?
Answer: 4

Explanation:
Given that,
You have 24 stones for the game mancala.
There are 6 holes on the board.
Each hole gets an equal number of stones.
24/6 = 4
Thus you put 4 stones in each hole.

Question 15.
Number Sense
Write the correct symbol to make each equation true

Answer:

Think and Grow: Modeling Real Life

You have 54 craft sticks. You use all of the sticks to make hexagons. How many hexagons can you make?

Division equation:
You can make _____ hexagons.

Answer:
Given,
You have 54 craft sticks. You use all of the sticks to make hexagons.
hexagon – 6
54/6 = 9
You can make 9 hexagons.

Show and Grow

Question 16.
You use 35 craft sticks to make 7 polygons. You use the same number of craft sticks for each polygon. How many craft sticks do you use for each polygon?
Answer: 5

Explanation:
Given,
You use 35 craft sticks to make 7 polygons. You use the same number of craft sticks for each polygon.
35/7 = 5
Thus you use 5 craft sticks for each polygon.

Question 17.
There are 42 students in gym class. The teacher divides the students into 7 teams. How many more students would be on each team if the teacher divides the students into 6 teams?
Answer:
Given that,
There are 42 students in gym class. The teacher divides the students into 7 teams.
42/7 = 6 students
If the teacher divides the students into 6 teams then,
42/6 = 7 students

Question 18.
You have a tray of 12 oatmeal bars. You keep 6 of them. How many bars can you give to each of your 6 friends?
Answer:
Given that,
You have a tray of 12 oatmeal bars. You keep 6 of them.
12/6 = 2
Thus you can give 2 bars to each of your 6 friends.

Divide by 6 or 7 Homework & Practice 4.5

Complete the model and find the quotient
Question 1.
Find 18 ÷ 6

18 ÷ 6 = _____
Answer: 3

Explanation:
Number of rows = 6
Number of counters = 18
Divide the number of counters by the number of rows
18/6 = 3
Thus there are 3 counters in each row

Question 2.
Find 35 ÷ 7

35 ÷ 7 = _____
Answer: 5

Explanation:
Number of rows = 7
Number of counters = 35
Divide the number of counters by the number of rows
35/7 = 5
Thus there are 5 counters in each row

Find the quotient
Question 3.
42 ÷ 6 = _____
Answer: 7

Explanation:
Divide the two numbers 42 and 6.
42/6 = 7
Thus the quotient is 7.

Question 4.
28 ÷ 7 = _____
Answer: 4

Explanation:
Divide the two numbers 28 and 7
28/7 = 4
Thus the quotient is 4.

Question 5.
54 ÷ 6 = _____
Answer: 9

Explanation:
Divide the two numbers 54 and 6.
54/6 = 9
Thus the quotient is 9.

Question 6.

Answer: 7

Explanation:
Divide the two numbers 49 and 7.
49/7 = 7
Thus the quotient is 7.

Question 7.

Answer: 10

Explanation:
Divide the two numbers 7 and 70
70/7 = 10
Thus the quotient is 10.

Question 8.

Answer: 5

Explanation:
Divide the two numbers 30 and 6.
30/6 = 5
Thus the quotient is 5.

Find the missing divisor.
Question 9.
36 ÷ ____ = 6
Answer: 9

Explanation:
Let the missing divisor be x.
36 ÷ x = 6
36/x = 6
36 = 6 × x
x = 36/6
x = 6
Thus the missing divisor is 6.

Question 10.
14 ÷ ____ = 7
Answer: 2

Explanation:
Let the missing divisor be y.
14 ÷ y = 7
14/y = 7
14 = 7 × y
y = 14/7
y = 2
Thus the missing divisor is 2.

Question 11.
60 ÷ _____ = 6
Answer: 10

Explanation:
Let the missing divisor be z.
60 ÷ z = 6
60/z = 6
60 = 6 × z
z = 60/6
z = 10
Thus the missing divisor is 10.

Question 12.
Number Sense
There are 7 continents. A scientist has 63 days to spend studying on all the continents. She wants to spend an equal number of days on each one. How many days can she spend on each continent?
Answer: 9

Explanation:
Given,
There are 7 continents. A scientist has 63 days to spend studying on all the continents.
She wants to spend an equal number of days on each one.
63/7 = 9 days
Thus she can spend 9 days on each continent.

Question 13.
Logic
I am an odd number. When you multiply me by 6, then divide the product by 3, the quotient is 10. What number am I?
Answer: 5

Explanation:
Let us consider 5 to be the odd number.
Now multiply by 6.
5 × 6 = 30
Now divide by 3.
30/3 = 10
So, the number is 5.

Question 14.
DIG DEEPER!
You deal 52 cards to 7 players. Does each player get the same number of cards? Explain.

Answer: No. Because 52 is not the multiple of 7.
Thus each player will not get the same number of cards.

Question 15.
Modeling Real Life
There are 70 students at a summer camp. A counselor divides the students into teams of 10. How many more teams would the counselor make if he divides the students into teams of 7?
Answer:
Given,
There are 70 students at a summer camp. A counselor divides the students into teams of 10.
If he divides into 7 teams then there will be 10 students in each team.
70/7 = 10

Question 16.
Modeling Real Life
There are 31 students in your class. Seven students are called to the nurse’s office to get their hearing checked. Your teacher divides the rest of the students into groups of 6. How many groups are there?
Answer:
Given that,
There are 31 students in your class. Seven students are called to the nurse’s office to get their hearing checked.
31 – 7 = 24
Your teacher divides the rest of the students into groups of 6.
24/6 = 4 groups
Thus there are 4 groups.

Review & Refresh

Find the product
Question 17.

Answer: 48

Explanation:
Multiply the two numbers 8 and 6.
8 × 6 = 48

Question 18.

Answer: 72

Explanation:
Multiply the two numbers 9 and 8.
9 × 8 = 72

Question 19.

Answer: 80

Explanation:
Multiply the two numbers 10 and 8.
10 × 8 = 80

Question 20.

Answer: 0

Explanation:
Multiply the two numbers 0 and 8.
Any number multiplied by 0 will be always 0.
Thus 8 × 0 = 0

Question 21.

Answer: 8

Explanation:
Multiply the two numbers 1 and 8.
Any number multiplied by 1 will be always the same number.
So, 1 × 8 = 8

Lesson 4.6 Divide by 8 or 9

Explore and Grow

Use repeated subtraction to find 48 ÷ 8.
Answer:
48 – 8 = 40
40 – 8 = 32
32 – 8 = 24
24 – 8 = 16
16 – 8 = 8
8 – 8 = 0
At sixth number we got 0.
So, 48/8 in repeated subtraction is 6.

Reasoning
How many times did you subtract 8 from 48? Does the quotient represent the number of groups or the size of the groups? Explain.
Answer: I subtracted 8 from 48 six times. Yes, the quotient represents the number of groups or size of the groups.

Think and Grow: Divide 8 or 9

Example
Find 40 ÷ 8.
Think: 8 times what number is 40?
8 × ____= 40

Answer: 5

Explanation:
Number of rows = 8
Number of counters = 40
Divide the number of counters by the number of rows
40/8 = 5
Thus there are 5 columns.
8 times 5 is 40.

Example
Find 54 ÷ 9.
Think: 9 times what number is 54?
9 × ____ = 54

Answer: 6

Explanation:
Number of rows = 9
Number of counters = 54
Divide the number of counters by the number of rows
54/9 = 6
Thus there are 6 columns.
9 times 6 is 54.

Show and Grow

Complete the model and find the quotient
Question 1.
24 ÷ 8 = ____

24 ÷ 8 = ____
Answer: 3

Explanation:
Number of rows = 8
Number of counters = 24
Divide the number of counters by the number of rows
24/8 = 3
There are 3 columns
8 times 3 is 24.

Question 2.
18 ÷ 9 = ____

18 ÷ 9 = ____
Answer: 2

Explanation:
Number of rows = 9
Number of counters = 18
Divide the number of counters by the number of rows
18/9 = 2
There are 2 columns
2 times 9 is 18.

Apply and Grow: Practice

Complete the model and find the quotient
Question 3.
Find 36 ÷ 9.

36 ÷ 9 = _____
Answer: 4

Explanation:
Number of rows = 9
Number of counters = 36
Divide the number of counters by the number of rows
36/9 = 4
There are 4 columns
4 times 9 is 36.

Question 4.
Find 32 ÷ 8.

32 ÷ 8 = ____
Answer: 4

Explanation:
Number of rows = 8
Number of counters = 32
Divide the number of counters by the number of rows
32/8 = 4
There are 4 columns
8 times 4 is 32.

Find the quotient
Question 5.
45 ÷ 9 = _____
Answer: 5

Explanation:
Divide the two numbers 45 and 9
45/9 = 5
Thus the quotient is 5.

Question 6.
56 ÷ 8 = _____
Answer: 7

Explanation:
Divide the two numbers 56 and 8.
56/8 = 7
Thus the quotient is 7.

Question 7.
90 ÷ 9 = ____
Answer: 10

Explanation:
Divide the two numbers 90 and 9.
90/9 = 10
Thus the quotient is 10.

Question 8.

Answer: 2

Explanation:
Divide the two numbers 9 and 18
18/9 = 2
Thus the quotient is 2.

Question 9.

Answer: 6

Explanation:
Divide the two numbers 8 and 48
48/8 = 6
Thus the quotient is 6.

Question 10.

Answer: 2

Explanation:
Divide the two numbers 16 and 8.
16/8 = 2
Thus the quotient is 2.

Question 11.
Divide 54 by 9.
Answer: 6

Explanation:
Divide the two numbers 54 and 9
54/9 = 6
Thus the quotient is 6.

Question 12.
Divide 64 by 8.
Answer: 8

Explanation:
Divide the two numbers 64 and 8.
64/8 = 8
Thus the quotient is 8.

Question 13.
Divide 63 by 9
Answer: 7

Explanation:
Divide the two numbers 63 and 9
63/9 = 7
Thus the quotient is 7.

Compare
Question 14.

Answer: =

Explanation:
Divide the two numbers
81 ÷ 9 = 9
72 ÷ 8 = 9
9 = 9
So, 81 ÷ 9 = 72 ­÷ 8

Question 15.

Answer: >

Explanation:
Divide the two numbers
72 ÷ 9 = 8
40 ÷ 8 = 5
8 > 5
So, 72 ÷ 9 > 40 ÷ 8

Question 16.

Answer: <

Explanation:
Divide the two numbers
16 ÷ 8 = 2
27 ÷ 9 = 3
2 < 3
So, 16 ÷ 8 < 27 ÷ 9

Question 17.
A comic book has 63 pages. You read 9 pages each night. How many nights will it take to read the entire book?
Answer: 7

Explanation:
Given that,
A comic book has 63 pages. You read 9 pages each night.
63/9 = 7
Thus it will take 7 nights to read the entire book.

Question 18.
Logic
Find the missing number.

Answer: 8

Explanation:
8 times 8 is 64.
Thus the missing number is 8.

Think and Grow: Modeling Real Life

There are 9 innings in a baseball game. The table shows how many innings Newton played each season. How many games has Newton played in all?

Division equation:
Newton played ______ games in all.

Answer:
Given,
There are 9 innings in a baseball game.
The table shows how many innings Newton played each season.
Season 1: 81/9 = 9 games
Season 2: 63/9 = 7 games
9 + 7 = 16 games
Therefore Newton played 16 games in all.

Show and Grow

Question 19.
The third-grade and fourth-grade classes are going on a field trip. A van can carry 8 students. How many vans are needed in all?

Answer:
Given that,
The third-grade and fourth-grade classes are going on a field trip.
A van can carry 8 students.
Grade 3 – 48 students
Grade 4 – 56 students
third grade – 48/8 = 6 vans
fourth grade – 56/8 = 7 vans
6 + 7 = 13 vans
Thus 13 vans are needed in all.

Question 20.
Seventy-two ballet dancers are arranged into an array with 9 columns. How many dancers are in each column?
Answer: 8 dancers

Explanation:
Given that,
Seventy-two ballet dancers are arranged into an array with 9 columns.
72/9 = 8
Thus 8 dancers are in each column.

Question 21
You make 15 paper elephants and 17 paper lions. You give all of the animals away to 8 friends. Each friend gets the same number of animals. How many animals does each friend get?

Answer: 4

Explanation:
Given that,
You make 15 paper elephants and 17 paper lions.
15 + 17 = 32
You give all of the animals away to 8 friends.
Each friend gets the same number of animals.
32/8 = 4
Thus each friend gets 4 animals.

Divide by 8 or 9 Homework & Practice 4.6

Complete the model and find the quotient
Question 1.
Find 16 ÷ 8.

16 ÷ 8 = ____
Answer: 2

Explanation:
Number of rows = 8
Number of counters = 16
Divide the number of counters by the number of rows
Division equation is 16 ÷ 8 = 2
Thus the number of columns is 2.

Question 2.
Find 45 ÷ 9.

45 ÷ 9 = _____
Answer: 5

Explanation:
Number of rows = 9
Number of counters = 45
Divide the number of counters by the number of rows
Division equation is 45 ÷ 9 = 5
Thus the number of columns is 5.

Find the quotient
Question 3.
48 ÷ 8. = _____
Answer: 6

Explanation:
Divide the two numbers 48 and 8.
48/8 = 6
Thus the quotient is 6.

Question 4.
63 ÷ 9 = ______
Answer: 7

Explanation:
Divide the two numbers 63 and 9.
63/9 = 7
Thus the quotient is 7.

Question 5.
54 ÷ 9 = _____
Answer: 6

Explanation:
Divide the two numbers 54 and 9.
54/9 = 6
Thus the quotient is 6.

Question 6.

Answer: 4

Explanation:
Divide the two numbers 32 and 8.
32/8 = 4
Thus the quotient is 4.

Question 7.

Answer: 9

Explanation:
Divide the two numbers 72 and 8.
72/8 = 9
Thus the quotient is 9.

Question 8.

Answer: 2

Explanation:
Divide the two numbers 18 and 9.
18/9 = 2
Thus the quotient is 2.

Compare
Question 9.

Answer: =

Explanation:
Divide the two numbers
90/9 = 10
80/8 = 10
10 = 10
90 ÷ 9 = 80 ÷ 8

Question 10.

Answer:  <

Explanation:
Divide the two numbers
27/9 = 3
56/8 = 7
3 < 7
27 ÷ 9 < 56 ÷ 8

Question 11.

Answer: >

Explanation:
Divide the two numbers
72 ÷ 9 = 8
40 ÷ 8 = 5
8 > 5
72 ÷ 9 > 40 ÷ 8

Question 12.
A food truck owner needs 64 whole wheat pitas. The pitas come in packages of 8. How many packages should she buy?
Answer: 8

Explanation:
Given that,
A food truck owner needs 64 whole-wheat pitas. The pitas come in packages of 8.
64 ÷ 8 = 8
Thus she should buy 8 packages.

Question 13.
Patterns
Complete the division tables.

Answer:

Question 14.
Structure
Describe two different ways to find 54 ÷ 9.
Answer: 6
The different ways to find 54 ÷ 9 is a division equation and multiplication equation.
54/9 = 6
6 × 9 = 54

Question 15.
Modeling Real Life
A youth group leader is preparing for a remote-control car race. Batteries are sold in packs of 8. How many packs of batteries should he buy?

Answer:
Given,
A youth group leader is preparing for a remote-control car race.
Batteries are sold in packs of 8.
cars – 48
48 ÷ 8 = 6 batteries for cars
24 ÷ 8 = 3 batteries for remote

Question 16.
Modeling Real Life
Newton divides 32 treats equally between 8 friends. Descartes divides 27 treats equally between 9 friends. Whose friends get more treats? Explain.
Answer:
Given that,
Newton divides 32 treats equally between 8 friends.
32 ÷ 8 = 4
Descartes divides 27 treats equally between 9 friends.
27 ÷ 9 = 3
Thus Newton’s friends get more treats.

Review & Refresh

Find the missing factor.
Question 17.
____ × 4 = 0
Answer: 0

Explanation:
Let the missing factor be x.
x × 4 = 0
x = 0/4
x = 0
Thus the missing factor is 0.

Question 18.
1 × ____ = 2
Answer: 2

Explanation:
Let the missing factor be y.
1 × y = 2
y = 2/1
y = 2
Thus the missing factor is 2.

Question 19.
9 × ____ = 9
Answer: 1

Explanation:
Let the missing factor be z.
9 × z = 9
z = 9/9
z = 1
Thus the missing factor is 1.

Lesson 4.7 Divide by 0 or 1

Explore and Grow

Find the quotients

Answer:

Structure
What patterns do you notice? Use the patterns to find each quotient.
52 ÷ 52 = _____
0 ÷ 52 = _____
52 ÷ 1 = ______
Answer:
52 ÷ 52 = 1
Any number divided by the same number will be always 1.
0 ÷ 52 = 0
Any number divided by 0 will be always 0.
52 ÷ 1 = 52
Any number divided by 1 will be the same number.

Think and Grow: Divide by 0 or 1

Dividing a number by 1 or itself:
• Any number divided by1 is itself.
• Any number (except 0) divided by itself is 1.
Example
Find 7 ÷ 1.
Think: 1 time what number is 7?
1 × ____ = 7
Write: 7 × 1 = _____

Answer:
Any number divided by 1 will be the same number.
7 × 1 = 7

Example
Find 5 ÷ 5.
Think: 5 times what number is 5?
5 × ____ = 5
Write: 5 ÷ 5 = _____

Dividing with 0:
• 0 divided by any number (except 0) is 0.
• You cannot divide by 0.

Answer:
5 × 1 = 5
5 times 1 is 5.
5 ÷ 5 = 1

Example
Find 0 ÷ 8.
Think: 8 times what number is 0?
8 × ____ = 0
Write: 0 × 8 = _____

Answer:
0 times 8 is 0.
Any number multiplied by 0 is always 0.
0 × 8 = 0

Example
Find 9 ÷ 0.
Think: 0 times what number is 9?
There is no such number. So, you cannot divide by 0.

Show and Grow

Write the related multiplication fact. Then find the quotient.
Question 1.
2 ÷ 2 = _____
Answer: 1

Explanation:
The related multiplication fact for 2, 1 is
2 × 1 = 2
1 × 2 = 2
2 ÷ 2 = 1
Thus the quotient is 1.

Question 2.
0 ÷ 3 = _____
Answer: 0

Explanation:
The related multiplication fact for 0, 3 is
0 × 3 = 0
3 × 0 = 0
Thus the quotient is 0.

Question 3.
10 ÷ 1 =_____
Answer: 10

Explanation:
The related multiplication fact for 1, 10
1 × 10 = 10
10 × 1 = 10
10 ÷ 1 = 10
Thus the quotient is 10.

Question 4.
25 ÷ 25 = _____
Answer: 1

Explanation:
The related multiplication fact for 25, 1
1 × 25 = 25
25 × 1 = 25
25 ÷ 25 = 1
Thus the quotient is 1.

Apply and Grow: Practice

Write the related multiplication fact. Then find the quotient.
Question 5.
6 ÷ 1 = ______
Answer: 6

Explanation:
The related multiplication fact for 6,1
1 × 6 =6
6 × 1 = 6
6 ÷ 1 = 6
Thus the quotient is 6.

Question 6.
0 ÷ 2 = _____
Answer: 0

Explanation:
The related multiplication fact for 0,2
Any number divided by 0 will be always 0.
0 ÷ 2 = 0
Thus the quotient is 0.

Question 7.
9 ÷ 9 = ____
Answer: 1

Explanation:
The related multiplication fact for 1, 9
1 × 9 = 9
9 × 1 = 9
9 ÷ 9 = 1
Thus the quotient is 1.

Question 8.
8 ÷ 1 = _____
Answer: 8

Explanation:
The related multiplication fact for 1, 8
1 × 8 = 8
8 × 1 =8
8 ÷ 1 = 8
Thus the quotient is 8.

Find the quotient
Question 9.
5 ÷ 1 = _____
Answer: 5

Explanation:
Any number divided by 1 will be the same number.
5/1 = 5
Thus the quotient is 5

Question 10.
0 ÷ 7 = _____
Answer: 0

Explanation:
Any number divided by 0 will be always 0.
0/7 = 0
Thus the quotient is 0.

Question 11.
4 ÷ 4 = _____
Answer: 1

Explanation:
4/4 = 1
Thus the quotient is 1.

Question 12.

Answer: 15

Explanation:
Any number divided by 1 will be the same number.
15/1 = 1
Thus the quotient is 15.

Question 13.

Answer: 0

Explanation:
Any number divided by 0 will be always 0.
0/6 = 0
Thus the quotient is 0.

Question 14.

Answer: 1

Explanation:
24/24 = 1
Thus the quotient is 1.

Find the missing dividend or divisor.
Question 15.
2 ÷ ____ = 2
Answer: 1

Explanation:
Let the missing divisor be x.
2 ÷ x = 2
2/x = 2
2 = 2 × x
x = 2/2
x = 1
Thus the missing divisor is 1.

Question 16.
____ ÷ 12 = 0
Answer: 0

Explanation:
Let the missing divisor be y.
y ÷ 12 = 0
y/12 = 0
y = 0 × 12
y = 0
Thus the missing divisor is 0.

Question 17.
8 ÷ ____ = 1
Answer: 8

Explanation:
Let the missing divisor be z.
8 ÷ z = 1
8/z = 1
8 = 1 × z
z = 8
Thus the missing divisor is 8.

Compare
Question 18.

Answer: >

Explanation:
Divide the two numbers
4 ÷ 1 = 4
3 ÷ 1 = 3
4 > 3
So, 4 ÷ 1 > 3 ÷ 1

Question 19.

Answer: =

Explanation:
Any number divided by 0 will be always 0.
0 ÷ 6 = 0
0 ÷ 9 = 0
So, 0 ÷ 6 = 0 ÷ 9

Question 20.

Answer: <

Explanation:
Any number divided by 0 will be always 0.
0 ÷ 5 = 0
7 ÷ 1 = 7
0 < 7
0 ÷ 5 < 7 ÷ 1

Question 21.
There are 2 cheese blocks. Each mousetrap has 1 cheese block on it. How many mousetraps are there?

Answer: 2

Explanation:
Given that,
There are 2 cheese blocks. Each mousetrap has 1 cheese block on it.
2 ÷ 1 = 2
Thus there are 2 mousetraps.

Question 22.
Reasoning
Your friend says 14 ÷ 0 and 0 ÷ 14 both equal 0. Is your friend correct? Explain.
Answer: Incorrect

Explanation:
Your friend is incorrect because 0 divided by any number except 0 will be always 0.
0/14 = 0
but
14/0 =  infinity

Think and Grow: Modeling Real Life

A clown shares 10 balloons equally with 10 children. How many balloons does each child receive?

Division equation:
Each child receives a _____ balloon.

Answer: 1

Explanation:
Given that,
A clown shares 10 balloons equally with 10 children.
10/10 = 1
Therefore each child receives 1 balloon.

Show and Grow

Question 23.
You have 9 tokens. You need 1 token to play an arcade game. How many games can you play?
Answer: 9

Explanation:
Given that,
You have 9 tokens. You need 1 token to play an arcade game.
9/1 = 9
Thus you can play 9 games.

Question 24.
You have 9 quarters. You put 5 of them in your backpack. You divide the other quarters equally among 4 friends. How many quarters does each friend get?
Answer: 1

Explanation:
Given,
You have 9 quarters. You put 5 of them in your backpack.
9 – 5 = 4
You divide the other quarters equally among 4 friends.
4/4 = 1
Thus each friend gets 1 quarter.

Question 25.
You ask your friend the question below.
“What is 475 divided by475?”
Your friend immediately says 1. How does your friend solve the problem so quickly? Explain.
Answer:
475/475 = 1
Your friend solved the problem very quickly because he used the logic ” Any number divided by the same number will be always 1″.

Divide by 0 or 1 Homework & Practice 4.7

Write the related multiplication fact. Then find the quotient.
Question 1.
3 ÷ 1 = _____
Answer: 3

Explanation:
The related multiplication fact for 1,3
1 × 3 = 3
3 × 1 = 3
Divide the two numbers
3 ÷ 1 = 3
Thus the quotient is

Question 2.
0 ÷ 7 = _____
Answer: 0

Explanation:
The related multiplication fact for 0, 7
0 × 7 = 0
7 × 0 = 0
Divide the two numbers 0, 7
0/7 = 0
Thus the quotient is 0

Find the quotient.
Question 3.
8 ÷ 1 = _____
Answer: 8

Explanation:
The related multiplication fact for 1, 8
8 × 1 = 8
1 × 8 = 8
Divide the two numbers 8, 1
8/1 = 8
Thus the quotient is 8

Question 4.
0 ÷ 2 = _____
Answer: 0

Explanation:
The related multiplication fact for 0, 2
0 × 2 = 0
2 × 0 = 0
Divide the two numbers
0/2 = 0
Thus the quotient is 0

Question 5.
7 ÷ 7 = _____
Answer: 1

Explanation:
The related multiplication fact for 7, 1
1 × 7 = 7
7 × 1 = 7
Divide the two numbers
7/7 = 1
Thus the quotient is 1

Question 6.

Answer: 1

Explanation:
The related multiplication fact for 1, 12
12 × 1 = 12
1 × 12 = 12
Divide the two numbers
12/12 = 1
Thus the quotient is 1

Question 7.

Answer: 4

Explanation:
The related multiplication fact for 1, 4
1 × 4 = 4
4 × 1 = 4
Divide the two numbers
4/1 = 4
Thus the quotient is 4

Question 8.

Answer: 0

Explanation:
The related multiplication fact for 0, 10
0 × 10 = 0
10 × 0 = 0
Divide the two numbers
0/10 = 0
Thus the quotient is 0

Find the missing dividend or divisor
Question 9.
____ ÷ 3 = 0
Answer: 0

Explanation:
Let the missing dividend be x
x ÷ 3 = 0
x/3 = 0
x = 0 × 3
x = 0
Thus the missing dividend is 0.