{"id":50461,"date":"2023-11-05T16:08:06","date_gmt":"2023-11-05T10:38:06","guid":{"rendered":"https:\/\/ccssmathanswers.com\/?p=50461"},"modified":"2023-11-10T12:32:48","modified_gmt":"2023-11-10T07:02:48","slug":"square-of-a-trinomial","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/square-of-a-trinomial\/","title":{"rendered":"Square of a Trinomial Expansion | Perfect Square Trinomial Definition, Formula & Solved Examples"},"content":{"rendered":"

Do you want to expand trinomials easily without any confusion and hassle? You should refer to this page. Here, we have explained how to expand the Square of a Trinomial<\/strong> and perfect square trinomial definition and formulas. Students who need more subject knowledge about square trinomials and solve any kind of trinomial expansions must go with this article completely. In this article, you will also get some worked-out examples on Square of a Trinomial and Perfect square trinomial. So, let’s continue your read and learn the concept of square trinomial.<\/p>\n

Perfect Square Trinomial Definition & Formula<\/h2>\n

An expression obtained from the square of the binomial equation is a perfect square trinomial. When the trinomial is in the form ax\u00b2 + bx + c<\/em> then it is said to be a perfect square, if and only if it meets the condition b\u00b2 = 4ac.<\/em><\/p>\n

The Perfect Square Trinomial Formula is as follows,<\/p>\n

(ax)\u00b2+2abx+b\u00b2 = (ax+b)\u00b2<\/strong>
\n(ax)\u00b2\u22122abx+b\u00b2 = (ax\u2212b)\u00b2<\/strong><\/p>\n

How to Expand the Square of a Trinomial?<\/h3>\n

Here, we are discussing the expansion of the square of a trinomial (a + b + c).<\/p>\n

Let (b + c) = x<\/p>\n

(i)<\/strong> Then (a + b + c)2<\/sup>\u00a0= (a + x)2<\/sup>\u00a0= a2<\/sup>\u00a0+ 2ax + x2<\/sup>
\n= a2<\/sup>\u00a0+ 2a (b + c) + (b + c)2<\/sup>
\n= a2<\/sup>\u00a0+ 2ab + 2ac + (b2<\/sup>\u00a0+ c2<\/sup>\u00a0+ 2bc)
\n= a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup>\u00a0+ 2ab + 2bc + 2ca<\/p>\n

Therefore, (a + b + c)2<\/sup>\u00a0= a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> + 2ab + 2bc + 2ca.<\/p>\n

(ii)<\/strong> (a – b – c)2<\/sup>\u00a0= [a + (-b) + (-c)]2<\/sup>
\n= a2<\/sup>\u00a0+ (-b2<\/sup>) + (-c2<\/sup>) + 2 (a) (-b) + 2 (-b) (-c) + 2 (-c) (a)
\n= a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup>\u00a0\u2013 2ab + 2bc \u2013 2ca<\/p>\n

Therefore, (a – b – c)2<\/sup>\u00a0= a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> \u2013 2ab + 2bc \u2013 2ca.<\/p>\n

(iii)<\/strong> (a + b – c)2<\/sup> = [a + b + (-c)]2<\/sup>
\n= a2<\/sup> + b2<\/sup> + (-c)2<\/sup> + 2ab + 2 (b) (-c) + 2 (-c) (a)
\n= a2<\/sup> + b2<\/sup> + c2<\/sup> + 2ab \u2013 2bc – 2ca<\/p>\n

Therefore, (a + b – c)2<\/sup> = a2<\/sup> + b2<\/sup> + c2<\/sup> + 2ab \u2013 2bc – 2ca.<\/p>\n

(iv) <\/strong>(a – b + c)2<\/sup> = [a + (- b) + c]2<\/sup><\/p>\n

= a2<\/sup> + (-b2<\/sup>) + c2<\/sup> + 2 (a) (-b) + 2 (-b) (-c) + 2 (c) (a)
\n= a2<\/sup> + b2<\/sup> + c2<\/sup> \u2013 2ab \u2013 2bc + 2ca<\/p>\n

Therefore, (a – b + c)2<\/sup> = a2<\/sup> + b2<\/sup> + c2<\/sup> \u2013 2ab \u2013 2bc + 2ca.<\/p>\n

Solved Examples on Square Trinomial<\/h3>\n

1. Expand (x+4y+6z)2<\/sup><\/p>\n

Solution:<\/strong><\/p>\n

Given trinomial expression is (x+4y+6z)2<\/sup><\/p>\n

We know that (a + b + c)2<\/sup>\u00a0= a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> + 2ab + 2bc + 2ca.<\/p>\n

Here a=1x, b=4y, c=6z<\/p>\n

Now, substitute the values in the expression of (a + b + c)2<\/sup><\/p>\n

Then (x+4y+6z)2<\/sup> = (1x)2<\/sup> + (4y)2<\/sup> + (6z)2<\/sup> + 2(1x)(4y) + 2(4y)(6z) + 2(6z)(1x)<\/p>\n

= x2<\/sup> + 16y2<\/sup> + 36z2<\/sup> + 8xy + 48yz + 12zx<\/p>\n

Hence, (x+4y+6z)2<\/sup> =\u00a0x2<\/sup> + 16y2<\/sup> + 36z2<\/sup> + 8xy + 48yz + 12zx.\u00a0<\/strong><\/p>\n

2. Is the trinomial x2\u00a0<\/sup>\u2013 6x + 9 a perfect square?<\/p>\n

Solution:<\/strong><\/p>\n

Given trinomial is x2<\/sup> \u2013 6x + 9, now calculate the expression and find it is a perfect square or not.<\/p>\n

x2<\/sup> \u2013 6x + 9 = x2<\/sup>\u00a0\u2013 3x \u2013 3x + 9
\n= x(x \u2013 3) \u2013 3(x \u2013 3)
\n= (x \u2013 3)(x \u2013 3)<\/p>\n

Otherwise,<\/p>\n

x2<\/sup>\u00a0\u2013 6x + 9 = x2<\/sup>\u00a0\u2013 2(3)(x) + 32 <\/sup>= (x \u2013 3)2<\/sup><\/p>\n

The factors of the given equation are a perfect square.<\/p>\n

Therefore, the given trinomial is a perfect square.<\/p>\n

3. Simplify a + b + c = 16 and ab + bc + ca = 40.\u00a0Find the value of a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup>.<\/p>\n

Solution:<\/strong><\/p>\n

As per the given question, a + b + c = 16<\/p>\n

Now, by squaring both sides, we get<\/p>\n

(a+ b + c)2<\/sup> = (16)2<\/sup><\/p>\n

a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> + 2ab + 2bc + 2ca = 256<\/p>\n

a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> + 2(ab + bc + ca) = 256<\/p>\n

a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> + 2 \u00d7 40 = 256 [Given, ab + bc + ca = 40]<\/p>\n

a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> + 80 = 256<\/p>\n

At this step, we have to subtract 80 from both sides<\/p>\n

a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> + 80 \u2013 80 = 256 \u2013 80<\/p>\n

a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> = 176<\/p>\n

Hence, the square of a trinomial formula will help us to expand and get the result for a2<\/sup>\u00a0+ b2<\/sup>\u00a0+ c2<\/sup> is 176.\u00a0<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Do you want to expand trinomials easily without any confusion and hassle? You should refer to this page. Here, we have explained how to expand the Square of a Trinomial and perfect square trinomial definition and formulas. Students who need more subject knowledge about square trinomials and solve any kind of trinomial expansions must go … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":50533,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[],"jetpack_featured_media_url":"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Square-of-a-Trinomial-.png","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/50461"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=50461"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/50461\/revisions"}],"predecessor-version":[{"id":92543,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/50461\/revisions\/92543"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media\/50533"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=50461"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=50461"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=50461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}