Factoring Difference of Squares Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Factoring Difference of Squares.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Recognizing and factoring expressions of the form a2−b2a^2 - b^2 into the product (a+b)(a−b)(a + b)(a - b).

When you multiply (a+b)(a−b)(a + b)(a - b), the middle terms cancel: a2−ab+ab−b2=a2−b2a^2 - ab + ab - b^2 = a^2 - b^2. So any time you see a perfect square minus a perfect square, you can instantly factor it. Think of it as a rectangle whose area is the difference of two square areas.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: When two perfect squares are subtracted, the expression factors into the conjugate pair (a+b)(a−b)(a+b)(a-b).

Common stuck point: The procedure for factoring difference of squares is the easy part; the trap is trying to factor a2+b2a^2+b^2 the same way. Asking "Are both terms perfect squares with a minus sign between them and nothing in the middle?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are both terms perfect squares with a minus sign between them and nothing in the middle?

Worked Examples

Example 1

easy
Factor x2−49x^2 - 49.

Answer

(x+7)(x−7)(x + 7)(x - 7)

First step

1
Step 1: Recognize the form a2−b2a^2 - b^2 where a=xa = x and b=7b = 7.

Full solution

  1. 2
    Step 2: Apply the formula: (a+b)(a−b)=(x+7)(x−7)(a+b)(a-b) = (x+7)(x-7).
  2. 3
    Step 3: Verify: (x+7)(x−7)=x2−7x+7x−49=x2−49(x+7)(x-7) = x^2 - 7x + 7x - 49 = x^2 - 49 ✓
The difference of squares pattern a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b) works because the middle terms cancel. Both terms must be perfect squares separated by subtraction.

Example 2

medium
Factor 16x2−25y216x^2 - 25y^2.

Example 3

medium
Factor 49x2−100y249x^2 - 100y^2.

Example 4

medium
Factor completely x4−16x^4 - 16.

Example 5

medium
Factor x6−64x^6 - 64 as a difference of two squares.

Example 6

medium
Factor (x+1)2−9(x + 1)^2 - 9.

Example 7

hard
Factor a2−(b+c)2a^2 - (b + c)^2.

Example 8

hard
Use difference of squares to compute (2025)2−(2024)2(2025)^2 - (2024)^2.

Example 9

hard
Factor x2−6x+9−y2x^2 - 6x + 9 - y^2 by grouping.

Example 10

hard
Factor (2x+1)2−(x−3)2(2x + 1)^2 - (x - 3)^2.

Example 11

challenge
Factor x8−1x^8 - 1 completely over the reals.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Factor x2−1x^2 - 1.

Example 2

hard
Factor x4−81x^4 - 81 completely.

Example 3

easy
Factor x2−9x^2 - 9.

Example 4

easy
Factor x2−25x^2 - 25.

Example 5

easy
Factor 4x2−14x^2 - 1.

Example 6

easy
Factor x2−49x^2 - 49.

Example 7

easy
Factor 9−x29 - x^2.

Example 8

easy
Factor x2−100x^2 - 100.

Example 9

easy
Factor 16−y216 - y^2.

Example 10

easy
Factor 25x2−3625x^2 - 36.

Example 11

medium
Factor completely: x4−16x^4 - 16.

Example 12

medium
Factor completely: 2x2−502x^2 - 50.

Example 13

medium
Factor completely: 48−3y248 - 3y^2.

Example 14

medium
Factor x2−14x^2 - \frac{1}{4}.

Example 15

medium
Factor 49a2−64b249a^2 - 64b^2.

Example 16

medium
Factor completely: x4−81x^4 - 81.

Example 17

medium
Use difference of squares to compute 532−47253^2 - 47^2 mentally.

Example 18

medium
Factor (x+1)2−9(x+1)^2 - 9.

Example 19

medium
Factor 100x2−49100x^2 - 49.

Example 20

challenge
Factor completely: x4−13x2+36x^4 - 13x^2 + 36.

Example 21

challenge
Factor x4+x2+1x^4 + x^2 + 1 completely over the integers.

Example 22

challenge
Factor completely: a4−b4a^4 - b^4.

Example 23

easy
Factor x2−16x^2 - 16.

Example 24

easy
Factor 4−x24 - x^2.

Example 25

easy
Factor x2−64x^2 - 64.

Example 26

easy
Factor 9x2−259x^2 - 25.

Example 27

easy
Factor 1−49y21 - 49y^2.

Example 28

medium
Factor 50x2−850x^2 - 8.

Example 29

medium
Factor x2−14x^2 - \tfrac{1}{4}.

Example 30

medium
Factor 25x2−125x^2 - 1.

Example 31

medium
Factor 36x2−49y236x^2 - 49y^2.

Example 32

medium
Solve x2−49=0x^2 - 49 = 0 by factoring.

Example 33

hard
Factor 16x4−81y416x^4 - 81y^4 completely.

Example 34

hard
Factor x29−y225\tfrac{x^2}{9} - \tfrac{y^2}{25}.

Example 35

hard
Solve 4x2−25=04x^2 - 25 = 0 by factoring.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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