Difference of Two Squares

≈ 25 min

A difference of two squares is a perfect square minus a perfect square, and it factorises instantly:

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

Why it works: expand $(a-b)(a+b) = a^2 + ab - ab - b^2 = a^2 - b^2$ — the middle terms cancel.

Worked example 1. $x^2 - 9 = x^2 - 3^2 = (x - 3)(x + 3)$

Worked example 2. $4x^2 - 25 = (2x)^2 - 5^2 = (2x - 5)(2x + 5)$

Recognition checklist: two terms ✓ subtraction between them ✓ each term a perfect square ✓. Note: $x^2 + 9$ (a sum of squares) does NOT factorise over the real numbers.

Factorise fully: $x^2 - 9$

Factorise fully: $4x^2 - 25$

Which expression is a difference of two squares?

$x^2 + 16$

$x^2 - 16$

$x^2 - 15$

$2x^2 - 9$