Maths Library

Solving Quadratic Equations: Year 13 maths

This topic covers finding solutions to quadratic equations algebraically and numerically. Students practise factorising monic and non-monic quadratics, using square roots, completing the square, and applying the quadratic formula.

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What this covers

Solving Quadratic Equations is taught in Year 13 (age 17 to 18, KS5) as a topic to introduce. It sits in the Algebra strand.

Worked example

Solve x^2 - 5x + 6 = 0

  1. Factorise the quadratic into two brackets: (x - 2)(x - 3) = 0
  2. Set each factor to zero: x - 2 = 0 or x - 3 = 0
  3. Solve each linear equation: x = 2 or x = 3

Answer: x = 2, x = 3

Difficulty scale

The sheet generator offers five difficulty settings for this topic.

  1. Pure quadratics of the form x^2 - c = 0 with integer square roots.
  2. Monic quadratics x^2 + bx + c = 0 factorisable over integers with positive roots.
  3. Monic quadratics with mixed signs or requiring rearrangement to equal zero.
  4. Non-monic quadratics ax^2 + bx + c = 0 factorisable over integers.
  5. Equations requiring the quadratic formula or completing the square, giving surd or rounded decimal roots.

Common mistakes

  • Forgetting the negative square root when solving x^2 = k, writing only one solution.
  • Failing to rearrange the quadratic into the standard form ax^2 + bx + c = 0 before factorising.

Curriculum links

Indicative matches to National Curriculum for England. Check wording and codes against the official document.

  • Solve non-linear simultaneous equations arising from the intersections of lines, curves, and circles.
  • Solve quadratic equations that emerge from trigonometric and exponential substitutions.

Sample sheet and answers

Six questions at the level's usual difficulty. Open the editor to change, add or regenerate any of them.

Solving Quadratic Equations

Year 13 maths

Qs
6
Marks
24
NameDateScore / 24
1.
Solve x2 − 1x − 2 = 0.
[4]
2.
Solve x2 + 9x + 18 = 0.
[4]
3.
Solve x2 + 8x + 16 = 0.
[4]
4.
Solve x2 − 3x − 4 = 0.
[4]
5.
Solve x2 − 49 = 0.
[4]
6.
Solve x2 + 10x + 21 = 0.
[4]

Answer key

  1. x = −1 or x = 2Factorise: (x + 1)(x − 2) = 0; So x = −1 or x = 2
  2. x = −6 or x = −3Factorise: (x + 6)(x + 3) = 0; So x = −6 or x = −3
  3. x = −4Factorise: (x + 4)(x + 4) = 0; So x = −4 or x = −4
  4. x = −1 or x = 4Factorise: (x + 1)(x − 4) = 0; So x = −1 or x = 4
  5. x = −7 or x = 7Factorise: (x + 7)(x − 7) = 0; So x = −7 or x = 7
  6. x = −7 or x = −3Factorise: (x + 7)(x + 3) = 0; So x = −7 or x = −3