The Difference of Two Squares

≈ 20 min

A very common pattern factorises instantly once you recognise it — the difference of two squares:

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

Why does it work? Expand the right side: $(a-b)(a+b) = a^2 + ab - ab - b^2 = a^2 - b^2$ . The middle terms cancel, leaving just the difference.

To use it, spot that both terms are perfect squares and there is a minus between them.

x^2 - 9 = (x - 3)(x + 3), \qquad 4x^2 - 25 = (2x - 5)(2x + 5)

In the second, $4x^2 = (2x)^2$ and $25 = 5^2$ , so $a = 2x$ and $b = 5$ .

Factorise $x^2 - 16$ .

$(x - 4)(x + 4)$

$(x - 8)(x + 8)$

$(x - 4)^2$

$(x - 16)(x + 1)$

Factorise $x^2 - 49$ . Answer in factored form, e.g. (x-7)(x+7).

Factorise $9x^2 - 25$ . Answer in factored form, e.g. (3x-5)(3x+5).

Which expression cannot be factorised as a difference of two squares?

$x^2 - 1$

$x^2 + 9$

$4x^2 - 9$

$x^2 - 100$