Quadratic equations

A quadratic has an x² term, like x² + 5x + 6 = 0. Solve it by factorising into two brackets, by the quadratic formula, or by completing the square. Most quadratics have two answers.

Key facts

  • Factorise x² + bx + c: find two numbers that multiply to c and add to b.
  • If two brackets multiply to 0, one of them must be 0.
  • Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a.
  • The answers are where the graph crosses the x-axis.

Worked examples

Example 1

Solve x² + 5x + 6 = 0.

  1. Two numbers that multiply to 6 and add to 5: 2 and 3. (x + 2)(x + 3) = 0
  2. One bracket must be zero. x = −2 or x = −3

Answer: x = −2 or x = −3

Example 2

Solve x² − 9 = 0.

  1. x² − 9 = (x + 3)(x − 3). (x + 3)(x − 3) = 0
  2. So x = −3 or x = 3

Answer: x = 3 or x = −3

Questions about quadratic equations