Developing more complex expressions

Identifying the correct identity

When faced with an expression to expand, you must first identify a and b, then choose the appropriate formula from the three identities covered previously.

Example with coefficients

Let’s expand (3x + 2y)². Here, a = 3x and b = 2y: (3x + 2y)² = (3x)² + 2 × (3x) × (2y) + (2y)² = 9x² + 12xy + 4y²

Example with a square root

The remarkable identities also work with square roots. Let’s expand (√x + 2)(√x – 2), for x ≥ 0: (√x + 2)(√x - 2) = (√x)^2 - 2^2 = x - 4

This method, known as the conjugate, is often used to simplify expressions containing square roots.

Example with a negative number

Let’s expand (5 - 2x)² with a = 5 and b = 2x: (5 - 2x)² = 25 - 2 × 5 × 2x + (2x)² = 25 - 20x + 4x²

Table of common cases

Expression	Identity used	Result
(3x+2y)^2	square of a sum	9x^2 + 12xy + 4y^2
(5-2x)^2	square of a difference	4x^2 - 20x + 25
(sqrt(x)+2)(sqrt(x)-2)	sum and difference	x - 4

Common pitfall

When b itself contains a coefficient (such as 2y or 2x), you must square the entire term: (2y)² = 4y², not 2y². Many pupils forget to square the coefficient at the same time as the variable.