Question

Prove the following expressions are trinomial perfect square:

x^2 + x + ¼​

Answer 1

Answer:

Step-by-step explanation:

Intro: Factoring perfect square trinomials

To expand any binomial, we can apply one of the following patterns.

   (a+b)2=a2+2ab+b2(\blueD a+\greenD b)^2=\blueD a^2+2\blueD a\greenD b+\greenD b^2(a+b)2=a2+2ab+b2left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared, equals, start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared

   (a−b)2=a2−2ab+b2(\blueD a-\greenD b)^2=\blueD a^2-2\blueD a\greenD b+\greenD b^2(a−b)2=a2−2ab+b2left parenthesis, start color #11accd, a, end color #11accd, minus, start color #1fab54, b, end color #1fab54, right parenthesis, squared, equals, start color #11accd, a, end color #11accd, squared, minus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared

[Where do these patterns come from?]

Note that in the patterns, aaaa and bbbb can be any algebraic expression. For example, suppose we want to expand (x+5)2(x+5)^2(x+5)2left parenthesis, x, plus, 5, right parenthesis, squared. In this case, a=x\blueD{a}=\blueD xa=xstart color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accd and b=5\greenD b=\greenD5b=5start color #1fab54, b, end color #1fab54, equals, start color #1fab54, 5, end color #1fab54, and so we get:

   (x+5)2=x2+2(x)(5)+(5)2=x2+10x+25\begin{aligned}(\blueD x+\greenD 5)^2&=\blueD x^2+2(\blueD x)(\greenD5)+(\greenD 5)^2\\\\ &=x^2+10x+25\end{aligned}(x+5)2​=x2+2(x)(5)+(5)2=x2+10x+25​

You can check this pattern by using multiplication to expand (x+5)2(x+5)^2(x+5)2left parenthesis, x, plus, 5, right parenthesis, squared. [I'd like to see this expansion, please!]

The reverse of this expansion process is a form of factoring. If we rewrite the equations in the reverse order, we will have patterns for factoring polynomials of the form a2±2ab+b2a^2\pm2ab+b^2a2±2ab+b2a, squared, plus minus, 2, a, b, plus, b, squared.

   a2+2ab+b2 =(a+b)2\blueD a^2+2\blueD a\greenD b+\greenD b^2~=(\blueD a+\greenD b)^2a2+2ab+b2 =(a+b)2start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared

   a2−2ab+b2 =(a−b)2\blueD a^2-2\blueD a\greenD b+\greenD b^2~=(\blueD a-\greenD b)^2a2−2ab+b2 =(a−b)2start color #11accd, a, end color #11accd, squared, minus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, minus, start color #1fab54, b, end color #1fab54, right parenthesis, squared

We can apply the first pattern to factor x2+10x+25x^2+10x+25x2+10x+25x, squared, plus, 10, x, plus, 25. Here we have a=x\blueD a=\blueD xa=xstart color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accd and b=5\greenD b=\greenD 5b=5start color #1fab54, b, end color #1fab54, equals, start color #1fab54, 5, end color #1fab54.

   x2+10x+25=x2+2(x)(5)+(5)2=(x+5)2\begin{aligned}x^2+10x+25&=\blueD x^2+2(\blueD x)(\greenD5)+(\greenD 5)^2\\\\ &=(\blueD x+\greenD 5)^2\end{aligned}x2+10x+25​=x2+2(x)(5)+(5)2=(x+5)2​

Expressions of this form are called perfect square trinomials. The name reflects the fact that this type of three termed polynomial can be expressed as a perfect square

Answer 2

Step-by-step explanation:

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