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All courses Math II · A-SSE.2 10 of 73
Use expression structure to identify useful rewrites, such as difference-of-squares factoring.

Factor a difference of squares

Problem
Factor x²−49 as a difference of squares. Enter the two square terms, identity, factorization, simplification if nested, and expansion check.
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Hint

Write each term as an explicit square.

a²−b²=(a−b)(a+b). Both terms must be squares and the operation must be subtraction.

Solution walkthrough

01

Recognize the square terms

\[x^2-49~=~x^2-7^2\]

Both terms are perfect squares, so the expression matches the difference-of-squares pattern.

02

Use the difference-of-squares formula

\[a^2-b^2~=~(a-b)(a+b)\]

Here \(a=x\) and \(b=7\).

03

Substitute the actual terms

\[x^2-7^2~=~(x-7)(x+7)\]

Replacing \(a\) and \(b\) gives the factorization.

04

State the factored form

\[\text{square}~\text{terms}=x²~\text{and}7²;~\text{identity}=a²−b²=(a−b)(a+b);~\text{factored}=(x−7)(x+7);~\text{expansion}~\text{check}=x²−49\]

This is the correct factorization of \(x^2-49\).

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Another way

  1. Expand \((x-7)(x+7)\) to check that the middle terms cancel and the product becomes \(x^2-49\).

!

Common mistake

Factoring it like a trinomial and looking for two numbers that add to \(0\) without first recognizing the difference-of-squares pattern.