What is important to know about Difference of two squares?

a² − b² = (a + b)(a − b) — this only works for subtraction, not addition.

What is important to know about Difference of two squares?

Both terms must be perfect squares.

What is important to know about Difference of two squares?

Can be used to factorise expressions like 9x² − 16 or x² − 25.

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Algebra · Higher

Difference of two squares

The difference of two squares is a factorisation pattern: a² − b² = (a + b)(a − b). Recognising this pattern allows quick factorisation without trial and error.

Revising difference of two squares the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Difference of two squares GCSE” on the site for the complete guide.

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Key facts to remember

1a² − b² = (a + b)(a − b) — this only works for subtraction, not addition.
2Both terms must be perfect squares.
3Can be used to factorise expressions like 9x² − 16 or x² − 25.
4Useful for mental arithmetic: e.g. 99 × 101 = (100 − 1)(100 + 1) = 10000 − 1 = 9999.
5Always look for a common factor before applying the difference of two squares.

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Formulas

Difference of two squares

a² − b² = (a + b)(a − b)

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Worked examples

Example 1

Factorise 25x² − 49

Working

Recognise 25x² = (5x)² and 49 = 7²
Apply a² − b² = (a + b)(a − b) with a = 5x, b = 7
= (5x + 7)(5x − 7)

Answer(5x + 7)(5x − 7)

Example 2

Factorise 2x² − 18

Working

First take out common factor of 2: 2(x² − 9)
Recognise x² − 9 = x² − 3²
= 2(x + 3)(x − 3)

Answer2(x + 3)(x − 3)

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Common mistakes

✗Trying to use this on a sum of squares: a² + b² cannot be factorised over the reals.

✗Forgetting to take out a common factor before applying the pattern.

✗Writing (a − b)(a − b) instead of (a + b)(a − b).

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Exam tips

✓Always check if both terms are perfect squares — if not, this technique does not apply.

✓Look for a common factor to remove first; then check for the difference of two squares.

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