Factorising quadratics (cross method)

2026-10-25Algebraic Manipulation Mr Ng

Factorise x² + bx + c and ax² + bx + c — a known weak area in S3 assessments.

Key formulas

x^{2} + (p+q)x + pq = (x+p)(x+q)

\text{For ax}^2 + bx + c \text{find factors of} a \times c \text{that sum to} b

When a = 1

Find two numbers that multiply to c and add to b. For x² − 7x + 6, the pair is −1 and −6 because (−1)+(−6)=−7 and (−1)(−6)=6, so (x−1)(x−6).

When a ≠ 1

Search for factor pairs of a×c whose signed sum is b, then split the middle term or use the cross method layout taught in class. Always take out a numerical common factor first if possible.

Perfect squares

x² + 12x + 36 = (x+6)². Recognising 36 = 6² and 12 = 2·6 saves time.

Exam habit

Expand your factors immediately to verify. If a ≠ 1, check the leading and constant terms first.

Worked example

Quadratic factors

Problem. Factorise x² − 7x + 6.

Solution. Need factors of 6 that add to −7: −1 and −6. So (x − 1)(x − 6).

Review quiz

1 / 4

x²+12x+36 =

Practice questions

Factorise x² + 12x + 36.
Factorise x² − 7x + 6.
Factorise x² + 5x − 14.
Factorise 2x² + 7x + 3.

Study notes

What if no integer factors work?

In junior papers the numbers are chosen to work; recheck signs of the constant term.

Should I take out 2 first from 2x² + 4x + 2?

Yes — 2(x² + 2x + 1) = 2(x+1)².