Expanding and factorising
Brackets, common factors, quadratics and the difference of two squares.
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 2.1.1Expand brackets and simplify algebraic expressions.
- 2.1.2Factorise expressions including quadratics and the difference of two squares.
6 minute read
Expanding and factorising
Expanding and factorising are opposite operations, and checking one with the other catches most errors.
Expanding
Multiply every term inside by the term outside. For two brackets, every term in the first multiplies every term in the second: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15 Watch signs carefully: (x − 4)(x + 2) = x² + 2x − 4x − 8 = x² − 2x − 8.
Factorising — always look for a common factor first
6x² + 9x = 3x(2x + 3). Take out the highest common factor, not just any factor.
Factorising quadratics of the form x² + bx + c
Find two numbers that multiply to c and add to b. For x² + 7x + 12: the numbers are 3 and 4, so it factorises to (x + 3)(x + 4). For x² − x − 6: multiply to −6, add to −1, giving −3 and +2, so (x − 3)(x + 2). When c is negative the two numbers have opposite signs; when c is positive they share the sign of b. Noticing that narrows the search immediately.
Difference of two squares
a² − b² = (a + b)(a − b) Recognise it whenever you see one square subtracted from another: x² − 25 = (x + 5)(x − 5), and 9x² − 16 = (3x + 4)(3x − 4). It appears constantly and is easy marks once spotted.
Think of it like this
Factorising is unmultiplying. Just as 12 can be rewritten as 3 × 4, a quadratic can be rewritten as two brackets — and expanding is how you check you unmultiplied correctly.
Worked examples
Method, step by step
Factorise x² − 3x − 10.
- 1Look for two numbers that multiply to −10 and add to −3.
- 2Since the product is negative, the numbers have opposite signs.
- 3−5 × 2 = −10 and −5 + 2 = −3, so the numbers are −5 and 2.
- 4Therefore x² − 3x − 10 = (x − 5)(x + 2).
- 5Check by expanding: x² + 2x − 5x − 10 = x² − 3x − 10 ✓
(x − 5)(x + 2)
Factorise 49x² − 64.
- 1Both terms are perfect squares: 49x² = (7x)² and 64 = 8².
- 2This is a difference of two squares, a² − b² = (a + b)(a − b).
- 3With a = 7x and b = 8, it factorises to (7x + 8)(7x − 8).
(7x + 8)(7x − 8)
Common misconceptions
- Expanding `(x + 3)²` as `x² + 9`. It means `(x + 3)(x + 3) = x² + 6x + 9` — the middle term is real.
- Losing a sign when the second bracket is subtracted. `−(x − 2)` is `−x + 2`, not `−x − 2`.
- Taking out only part of the common factor, e.g. writing `2(3x² + 4.5x)` instead of `3x(2x + 3)`.
In the exam
- Always check a factorisation by expanding it back. It takes seconds and catches sign errors reliably.
- Before anything else, check for a common factor — many quadratics become far easier once it is removed.