Factorising perfect squares

Read the identity backwards to factor. Look at x² + 4x + 4:

• x² = (x)²;
• 4 = 2²;
• 4x = 2(x)(2), twice their product.

So x² + 4x + 4 = (x + 2)².

Make friends with the square terms first. Find what the two end terms are squares of, then check that the middle is twice their product. 36x² + 12x + 1 = (6x)² + 2(6x)(1) + 1² = (6x + 1)².

Not a neat square? Take out a common factor. 50p² is not a perfect square, but 2 divides every term of 50p² + 60pq + 18q²: 2(25p² + 30pq + 9q²) = 2(5p + 3q)².

The minus version. Put –b in place of b: (a – b)² = a² – 2ab + b². The squares stay positive; only the middle term carries the minus. A minus middle term means a minus inside the bracket.

Squares just below a round number: Example 8: 29² = (30 – 1)² = 900 – 60 + 1 = 841.

Always check a factorisation by expanding it back.