Expanding and factorising

Expanding single brackets

Multiply each term inside the bracket by the term outside.

Example: 3(2x − 5) = 6x − 15.

Watch the signs carefully when the term outside is negative: −4(x − 7) = −4x + 28.

Expanding two brackets (FOIL or grid)

(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15.

CCEA accepts FOIL (First, Outer, Inner, Last) or a 2×2 grid — both score the M1 for showing all four products and A1 for the simplified answer.

Squaring a bracket — common mistake

(x + 4)² ≠ x² + 16. Write it out fully: (x + 4)(x + 4) = x² + 4x + 4x + 16 = x² + 8x + 16.

Factorising — common factor

Find the highest common factor (HCF) of all terms.

Example: 6x²y + 9xy² → HCF = 3xy → 3xy(2x + 3y).

Factorising quadratics x² + bx + c

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

Example: x² + 7x + 12. Factor pairs of 12: (1,12), (2,6), (3,4). 3 + 4 = 7 ✓. So x² + 7x + 12 = (x + 3)(x + 4).

For x² + bx − c: numbers have opposite signs.

Difference of two squares

a² − b² = (a − b)(a + b).

Example: 9x² − 25 = (3x − 5)(3x + 5). Recognise the form: two squared terms with a minus.

Factorising ax² + bx + c (Higher tier)

Method: find two numbers that multiply to a × c and add to b. Split the middle term, then factor by grouping.

Example: 2x² + 7x + 3. a×c = 6, sum 7 → 6 and 1. 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

Common CCEA exam tip

After factorising, expand mentally to check — if you don't get back to the original quadratic, your factorisation is wrong.