GCSE Revision

Quadratic formula and completing the square

M2 formulae

The quadratic formula (Higher tier only)

The solves any ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / 2a

It's on the formulae sheet. ± means work it out once with + and once with −.

  1. Rearrange to = 0 and write down a, b and c, with their signs.
  2. Work out b² − 4ac, then both answers.

Example: solve 2x² − 5x − 1 = 0. Give your answers to 2 decimal places.

  1. a = 2, b = −5, c = −1
  2. b² − 4ac = 25 − (4 × 2 × −1) = 25 + 8 = 33
  3. x = (5 ± √33) / 4 = 2.686… or −0.186…
  4. x = 2.69 or x = −0.19

Common mistake

If b = −5, then −b = 5 and b² = 25, not −25. On a calculator, type (−5)².

Key terms

Answers as surds (Higher tier only)

"Give your answer in surd form" means leave the square root in and simplify it.

Example: solve x² + 4x − 1 = 0, giving your answers in surd form.

  1. a = 1, b = 4, c = −1
  2. b² − 4ac = 16 − (4 × 1 × −1) = 16 + 4 = 20
  3. x = (−4 ± √20) / 2
  4. √20 = √4 × √5 = 2√5, so x = (−4 ± 2√5) / 2
  5. Divide each term by 2: x = −2 ± √5

The b² − 4ac tells you how many solutions there are: two if it's positive, one if it's 0, and none if it's negative, because you can't square root a negative number.

Completing the square (Higher tier only)

writes x² + bx + c as (x + p)² + q.

  1. Halve b to get p, and write (x + p)².
  2. Subtract p², because (x + p)² makes an extra p².
  3. Add c and simplify.

Example: write x² + 6x + 5 in the form (x + p)² + q.

  1. Half of 6 is 3: (x + 3)²
  2. (x + 3)² = x² + 6x + 9, which is 9 too many: (x + 3)² − 9
  3. (x + 3)² − 9 + 5 = (x + 3)² − 4

Harder example: complete the square for x² − 5x + 2.

  1. Half of −5 is −5/2: (x − 5/2)² − 25/4
  2. −25/4 + 2 = −25/4 + 8/4 = −17/4
  3. (x − 5/2)² − 17/4

Remember

Halve the b, then take away its square.

Key terms

Completing the square when a is more than 1 (Higher tier only)

Take a out of the x² and x terms first, then complete the square inside the bracket.

  1. Factorise a out of the first two terms.
  2. Complete the square inside the bracket.
  3. Multiply out the outer bracket and simplify.

Example: write 2x² + 12x + 7 in the form a(x + p)² + q.

  1. 2x² + 12x + 7 = 2(x² + 6x) + 7
  2. x² + 6x = (x + 3)² − 9, so 2[(x + 3)² − 9] + 7
  3. 2(x + 3)² − 18 + 7 = 2(x + 3)² − 11

Check by expanding: 2(x² + 6x + 9) − 11 = 2x² + 12x + 18 − 11 = 2x² + 12x + 7.

Solving by completing the square (Higher tier only)

Once a quadratic equation is in completed-square form, you can solve it with inverse operations.

  1. Complete the square.
  2. Get the squared bracket on its own.
  3. Square root both sides, with ±.
  4. Solve for x.

Example: solve x² + 6x + 5 = 0 by completing the square.

  1. (x + 3)² − 4 = 0
  2. (x + 3)² = 4
  3. x + 3 = ±2
  4. x = −1 or x = −5

Harder example: solve x² − 4x − 3 = 0, leaving your answers as surds.

  1. (x − 2)² − 4 − 3 = 0, so (x − 2)² − 7 = 0
  2. (x − 2)² = 7
  3. x − 2 = ±√7
  4. x = 2 + √7 or x = 2 − √7

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