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.

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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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