What is the key idea in difference of squares?

A difference of squares has two perfect-square terms separated by subtraction: a^2-b^2=(a-b)(a+b). The pattern does not apply to a sum of squares over the real numbers.

How should I practice difference of squares?

Study one worked example, cover the solution, reproduce it independently, and then complete mixed practice while checking each result.

Algebra · Step-by-step guide

Difference of Squares

Recognize and factor the difference-of-squares pattern, including expressions with common factors.

By Tyler Blovat·Updated September 22, 2026·9 min read

The short answer

Start with the central idea

A difference of squares has two perfect-square terms separated by subtraction: $\displaystyle a^2-b^2=(a-b)(a+b)$ . The pattern does not apply to a sum of squares over the real numbers.

\displaystyle a^2-b^2=(a-b)(a+b)

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

Check all three conditions

There must be two terms, subtraction between them, and each term must be a perfect square.

Remove a GCF first

Factoring out the greatest common factor may reveal a difference of squares inside.

Factor completely

After applying the pattern, inspect the factors again for another difference of squares.

See the structure

A picture makes the relationships easier to remember.

The coordinate graph marks the vertex and the two x-intercepts of a representative quadratic.

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Basic pattern

Factor $\displaystyle x^2-49$ .

Recognize $\displaystyle x^2$ and $\displaystyle 7^2$ .
Apply $\displaystyle a^2-b^2$ .
Use conjugate binomials.

Answer $\displaystyle (x-7)(x+7)$

Example 2With coefficients

Factor $\displaystyle 9y^2-16$ .

Write $\displaystyle 9y^2=(3y)^2$ .
Write $\displaystyle 16=4^2$ .
Apply the pattern.

Answer $\displaystyle (3y-4)(3y+4)$

Example 3Factor completely

Factor $\displaystyle 2x^4-32$ .

Take out 2: $\displaystyle 2(x^4-16)$ .
Factor $\displaystyle x^4-16=(x^2-4)(x^2+4)$ .
Factor $\displaystyle x^2-4$ again.

Answer $\displaystyle 2(x-2)(x+2)(x^2+4)$

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

Trying to factor a² + b² with this real-number pattern.
Forgetting to take out the GCF.
Stopping while a factor is still a difference of squares.

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Difference of Squares

Recognize the pattern and practice factoring completely.

Preview the worksheet

✓

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

Factor $\displaystyle x^2-81$ .

Show hint

Write 81 as a square.

Show worked solution

$\displaystyle 81=9^2$ .
Apply $\displaystyle a^2-b^2=(a-b)(a+b)$ .

Answer: $\displaystyle (x-9)(x+9)$

Factor $\displaystyle 25y^2-4$ .

Show hint

Identify both square roots.

Show worked solution

$\displaystyle 25y^2=(5y)^2$ and $\displaystyle 4=2^2$ .
Use conjugate factors.

Answer: $\displaystyle (5y-2)(5y+2)$

Factor completely: $\displaystyle 3x^2-48$ .

Show hint

Take out the GCF first.

Show worked solution

$\displaystyle 3x^2-48=3(x^2-16)$ .
$\displaystyle x^2-16=(x-4)(x+4)$ .

Answer: $\displaystyle 3(x-4)(x+4)$

Factor completely: $\displaystyle x^4-16$ .

Show hint

Treat $\displaystyle x^4$ as $\displaystyle (x^2)^2$ .

Show worked solution

$\displaystyle x^4-16=(x^2-4)(x^2+4)$ .
Factor $\displaystyle x^2-4$ again.

Answer: $\displaystyle (x-2)(x+2)(x^2+4)$

Can $\displaystyle x^2+36$ be factored by the difference-of-squares rule?

Show hint

Check the operation between terms.

Show worked solution

Both terms are squares, but they are added.
The real-number difference-of-squares pattern requires subtraction.

Answer: No; it is a sum of squares.

Frequently asked questions

Quick answers before you move on.

What should I remember first?

A difference of squares has two perfect-square terms separated by subtraction: $\displaystyle a^2-b^2=(a-b)(a+b)$ . The pattern does not apply to a sum of squares over the real numbers.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

Start with the central idea

How it works

Check all three conditions

Remove a GCF first

Factor completely

See the structure

Worked examples

Factor x2−49\displaystyle x^2-49x2−49.

Factor 9y2−16\displaystyle 9y^2-169y2−16.

Factor 2x4−32\displaystyle 2x^4-322x4−32.

Common mistakes—and how to avoid them

Turn the explanation into a skill

Difference of Squares

Practice problems with worked solutions

Frequently asked questions

Factoring Trinomials

Completing the Square

The Quadratic Formula

Factor $\displaystyle x^2-49$ .

Factor $\displaystyle 9y^2-16$ .

Factor $\displaystyle 2x^4-32$ .