GCSE Revision

Factorising quadratics

Factorising x² + bx + c

A expression has an x² term and no higher power, such as x² + 7x + 12. To factorise one, write it as two brackets.

  1. Find two numbers that multiply to give c (the number on its own).
  2. Of those, pick the pair that adds to give b (the number of x's).
  3. Write (x + first number)(x + second number).
  4. Check by expanding.

Example: factorise x² + 7x + 12.

  1. Pairs that multiply to 12: 1 and 12, 2 and 6, 3 and 4
  2. 3 + 4 = 7
  3. (x + 3)(x + 4)

Check: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12.

Key terms

The four sign cases

If c is positive, both brackets have the same sign as b. If c is negative, the signs are different, and the bigger number takes the sign of b.

ExpressionFactorised
x² + 7x + 12(x + 3)(x + 4)
x² − 7x + 12(x − 3)(x − 4)
x² + x − 12(x + 4)(x − 3)
x² − x − 12(x − 4)(x + 3)

Example: factorise x² − 2x − 15.

  1. c is negative, so the signs differ; b is negative, so the bigger number is negative.
  2. 3 × 5 = 15, and −5 + 3 = −2
  3. (x − 5)(x + 3)

Common mistake

Check the middle term when you expand: (x + 5)(x − 3) gives +2x, not −2x.

Factorising ax² + bx + c (Higher tier only)

When x² has a number in front, (after taking out any common factor).

  1. Find two numbers that multiply to a × c and add to b.
  2. Use them to split bx into two terms.
  3. Factorise each pair of terms, then take out the shared bracket.

Example: factorise 2x² + 7x + 3.

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

Harder example: factorise 3x² − 10x − 8.

  1. a × c = −24, and −12 × 2 = −24, −12 + 2 = −10
  2. 3x² − 12x + 2x − 8
  3. 3x(x − 4) + 2(x − 4) = (3x + 2)(x − 4)

The difference of two squares

A is one square number or term take away another. It factorises straight away:

a² − b² = (a + b)(a − b)

  1. Check both terms are squares and there's a minus between them.
  2. Square root each term.
  3. Write (first + second)(first − second).

Example: factorise x² − 25.

  1. x² = x × x and 25 = 5 × 5
  2. (x + 5)(x − 5)

Check: (x + 5)(x − 5) = x² − 5x + 5x − 25 = x² − 25. The middle terms cancel, which is why there's no x term.

Remember

Spot it by the shape: two squares and a minus sign, with no x term. x² + 25 doesn't factorise.

Key terms

Harder differences of two squares (Higher tier only)

The terms can have numbers in front, as long as each is a square.

Example: factorise 4x² − 9.

  1. 4x² = (2x)² and 9 = 3²
  2. (2x + 3)(2x − 3)

Harder example: factorise 3x² − 48 fully.

  1. 3x² and 48 aren't squares, but 3 is a common factor: 3(x² − 16)
  2. x² − 16 = (x + 4)(x − 4)
  3. 3(x + 4)(x − 4)

Check: 3(x² − 16) = 3x² − 48.

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