What you'll learn
Factorising a quadratic expression means writing it as a product of two brackets — the reverse of expanding. In this guide you will learn how to factorise simple quadratics of the form x² + bx + c, how to factorise harder quadratics where the x² coefficient is greater than 1, how to recognise and use the difference of two squares, and how factorising helps solve quadratic equations. This is one of the most important algebra skills at GCSE.
Key terms and definitions
Factorise — to write an expression as a product of factors (often two brackets).
Quadratic expression — an expression of the form ax² + bx + c.
Difference of two squares — an expression a² − b², which factorises to (a + b)(a − b).
Coefficient — the number multiplying a term (a is the coefficient of x²).
Factor pair — two numbers that multiply to give a particular product.
Core concepts
Factorising x² + bx + c
For a quadratic where the x² coefficient is 1, find two numbers that multiply to c and add to b. Then write the brackets using those numbers. For example, for x² + 7x + 12, two numbers that multiply to 12 and add to 7 are 3 and 4, so it factorises to (x + 3)(x + 4).
Dealing with signs
The signs of the numbers depend on b and c. If c is positive, both numbers have the same sign as b. If c is negative, the numbers have opposite signs. For example, x² − 2x − 15 = (x − 5)(x + 3): the numbers −5 and +3 multiply to −15 and add to −2.
Factorising ax² + bx + c (a ≠ 1)
When the x² coefficient is greater than 1, find two numbers that multiply to a × c and add to b, then split the middle term and factorise by grouping. For example, for 2x² + 7x + 3: a × c = 6, and 6 and 1 multiply to 6 and add to 7. Split: 2x² + 6x + 1x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Difference of two squares
An expression of the form a² − b² factorises to (a + b)(a − b). For example, x² − 25 = (x + 5)(x − 5), and 9x² − 16 = (3x + 4)(3x − 4). Look for two perfect squares separated by a minus sign, with no middle term.
Common factors first
Before factorising into brackets, always check for a common factor to take outside. For example, 2x² + 8x + 6 = 2(x² + 4x + 3) = 2(x + 1)(x + 3). Taking out the common factor first makes the numbers smaller and easier.
Worked examples
Example 1: Simple quadratic
Factorise x² + 9x + 20.
Two numbers multiplying to 20 and adding to 9 are 4 and 5: (x + 4)(x + 5).
Example 2: Difference of two squares
Factorise x² − 49.
This is a difference of two squares: x² − 7², so (x + 7)(x − 7).
Example 3: Harder quadratic
Factorise 3x² + 10x + 8.
a × c = 24; numbers 6 and 4 multiply to 24, add to 10. Split: 3x² + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2).
Common mistakes and how to avoid them
Wrong signs in the brackets. Use the rule: c positive → same signs; c negative → opposite signs.
Forgetting common factors. Always check for a common factor first.
Missing the difference of two squares. Two squares with a minus and no middle term factorise as (a + b)(a − b).
Errors with a ≠ 1. Multiply a × c, find the factor pair, split the middle term, then group.
Not checking by expanding. Multiply your brackets back out to confirm you get the original.
Exam technique for Factorising Quadratics
Take out common factors first to simplify.
Use the multiply-to-c, add-to-b rule for x² + bx + c, with correct signs.
Use a × c and grouping for harder quadratics.
Spot the difference of two squares instantly.
Check by expanding your answer to make sure it matches.
Quick revision summary
Factorising a quadratic reverses expanding, writing it as two brackets. For x² + bx + c, find two numbers that multiply to c and add to b ((x² + 7x + 12) = (x + 3)(x + 4)); if c is positive the numbers share b's sign, and if c is negative they have opposite signs. For ax² + bx + c with a > 1, find numbers multiplying to a × c and adding to b, split the middle term, and factorise by grouping (3x² + 10x + 8 = (3x + 4)(x + 2)). The difference of two squares, a² − b², always factorises to (a + b)(a − b) (x² − 25 = (x + 5)(x − 5)). Always take out common factors first to keep numbers small, and check by expanding. These skills are the gateway to solving quadratic equations by factorising. Take out common factors, apply the multiply/add rule with correct signs, use grouping for harder cases, recognise the difference of two squares, and verify by multiplying back.