Expanding brackets and factorisation

Algebra manipulation is tested on every OCR J560 paper. Expanding and factorising quadratics are higher-tier topics but appear on both tiers in the context of solving equations.

Expanding single brackets

Multiply each term inside the bracket by the term outside.

3(2x − 5) = 6x − 15. −2(3x + 4) = −6x − 8 (be careful with negatives).

Expanding double brackets (FOIL / grid method)

(a + b)(c + d) = ac + ad + bc + bd.

Example: (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15.

Grid method is equally valid:

	x	−5
x	x²	−5x
3	3x	−15

Sum: x² − 5x + 3x − 15 = x² − 2x − 15.

Difference of two squares

(a + b)(a − b) = a² − b².

Example: (x + 7)(x − 7) = x² − 49. No middle term!

Squaring a bracket

(x + a)² = x² + 2ax + a². Note: (x + 3)² ≠ x² + 9. The middle term 2 × 3 × x = 6x is essential.

Factorising

Common factor

6x² − 9x = 3x(2x − 3). Always look for the highest common factor first.

Factorising quadratics: ax² + bx + c where a = 1

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

Example: x² + 5x + 6 → numbers multiply to 6, add to 5 → (2, 3) → (x + 2)(x + 3).

Example: x² − 3x − 10 → multiply to −10, add to −3 → (2, −5) → (x + 2)(x − 5).

Factorising quadratics: ax² + bx + c where a ≠ 1

Find two numbers that multiply to a×c and add to b; split the middle term.

Example: 6x² + 11x + 3. a×c = 18; numbers that multiply to 18 and add to 11: (2, 9). 6x² + 2x + 9x + 3 = 2x(3x + 1) + 3(3x + 1) = (2x + 3)(3x + 1).

Difference of two squares factorisation

x² − 25 = (x + 5)(x − 5). a² − b² = (a + b)(a − b).

Common OCR exam mistakes

1. (x + 3)² = x² + 6x + 9 — NOT x² + 9. The middle term is always 2 × a × b.
2. Factorising a negative quadratic: −x² + 5x − 6 → factor out −1 first: −(x² − 5x + 6) = −(x − 2)(x − 3).
3. Giving a factorised answer when the question says "solve" — finish by setting each bracket to zero.