Expanding and Factorising

Expanding single brackets

Multiply every term inside by the term outside:

3(x + 4)      = 3x + 12
2x(x − 5)     = 2x² − 10x
−4(2x − 3)    = −8x + 12     (watch the signs!)

Expanding double brackets — FOIL

Multiply First, Outer, Inner, Last:

(x + 3)(x + 2)
= x·x + x·2 + 3·x + 3·2
= x² + 2x + 3x + 6
= x² + 5x + 6

Difference of two squares: (x + 5)(x − 5) = x² − 25.

Factorising — single bracket

Take out the Highest Common Factor (HCF):

6x + 9    = 3(2x + 3)
8x² − 12x = 4x(2x − 3)

Factorising quadratics x² + bx + c

Find two numbers that multiply to c and add to b:

x² + 7x + 10   →  +5 and +2  →  (x + 5)(x + 2)
x² − x − 6     →  −3 and +2  →  (x − 3)(x + 2)
x² − 9x + 20   →  −5 and −4  →  (x − 5)(x − 4)

Difference of two squares

a² − b² = (a + b)(a − b) → x² − 16 = (x + 4)(x − 4)

Common mistakes

• Forgetting to multiply the last terms in FOIL.
• Sign errors when c is negative (the numbers have opposite signs).

Exam tip

Always check by expanding back. If b is negative but c is positive, both numbers are negative.