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Algebra · Higher

Quadratic equations (factorising)

Quadratic equations have the form ax² + bx + c = 0. When the quadratic can be factorised, you set each bracket to zero to find the solutions (roots).

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Key facts to remember

  • 1Rearrange to ax² + bx + c = 0 before factorising.
  • 2Factorise into two brackets, then set each bracket = 0.
  • 3A quadratic can have 0, 1 or 2 solutions.
  • 4If the equation is x² = k, then x = ±√k.
  • 5For ax² + bx + c where a ≠ 1, find factors of ac that sum to b.
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Worked examples

Example 1

Solve x² + x − 6 = 0

Working

  1. Find factors of −6 that sum to +1: +3 and −2
  2. (x + 3)(x − 2) = 0
  3. x + 3 = 0 → x = −3
  4. x − 2 = 0 → x = 2
Answerx = −3 or x = 2
Example 2

Solve 2x² + 5x − 3 = 0

Working

  1. a × c = 2 × (−3) = −6
  2. Find factors of −6 that sum to 5: +6 and −1
  3. 2x² + 6x − x − 3 = 0
  4. 2x(x + 3) − 1(x + 3) = 0
  5. (2x − 1)(x + 3) = 0
  6. x = 1/2 or x = −3
Answerx = 1/2 or x = −3
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Common mistakes

✗Not rearranging to = 0 first (e.g. trying to factorise x² + x = 6).
✗Only giving one solution — quadratics almost always have two.
✗Setting ax² = 0 to get x = 0 instead of factorising properly.
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Exam tips

✓Always rearrange so one side is zero before factorising.
✓Check your solutions by substituting both back into the original equation.

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