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AQASpec A18Foundation & Higher~35 min

Solving Quadratic Equations

Mathematics · Topic revision workspace

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Topic overview

What you need to know

Solving Quadratic Equations is a key part of GCSE Mathematics. Build fluency with the method, then apply it to unfamiliar and multi-step problems. An equation of the form ax² + bx + c = 0 where a ≠ 0. The highest power of x is 2. Write ax² + bx + c = 0 as (x + p)(x + q) = 0, then set each bracket equal to zero. x = (−b ± √(b² − 4ac)) / 2a. Used to solve any quadratic equation ax² + bx + c = 0.

Exam tip

Write every stage of your method. Method marks can still be earned when the final answer is wrong.

Revision notes

Core ideas

Secure these ideas first. Say each definition in your own words, then connect it to the topic overview.

  • Quadratic Equation: An equation of the form ax² + bx + c = 0 where a ≠ 0. The highest power of x is 2.
  • Factorising to Solve: Write ax² + bx + c = 0 as (x + p)(x + q) = 0, then set each bracket equal to zero.
  • Quadratic Formula: x = (−b ± √(b² − 4ac)) / 2a. Used to solve any quadratic equation ax² + bx + c = 0.
  • Discriminant: b² − 4ac. It determines the number of real roots: positive = 2 roots, zero = 1 root, negative = no real roots.

Revision notes

Apply it in the exam

The exam will rarely ask for an isolated definition. Practise selecting the right idea and using it as part of a complete explanation or method.

  • Completing the Square: Write x² + bx + c in the form (x + b/2)² − (b/2)² + c. Used to solve quadratics and find the vertex.
  • Difference of Two Squares: a² − b² = (a + b)(a − b). A special factorisation pattern.
  • Parabola: The U-shaped curve produced when graphing a quadratic function. Opens upward if a > 0, downward if a < 0.
  • Roots / Solutions: The values of x where the quadratic equals zero. Graphically, these are the x-intercepts of the parabola.

Core knowledge

Key facts for Solving Quadratic Equations

1

Quadratic Equation

An equation of the form ax² + bx + c = 0 where a ≠ 0. The highest power of x is 2.

2

Factorising to Solve

Write ax² + bx + c = 0 as (x + p)(x + q) = 0, then set each bracket equal to zero.

3

Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a. Used to solve any quadratic equation ax² + bx + c = 0.

4

Discriminant

b² − 4ac. It determines the number of real roots: positive = 2 roots, zero = 1 root, negative = no real roots.

5

Completing the Square

Write x² + bx + c in the form (x + b/2)² − (b/2)² + c. Used to solve quadratics and find the vertex.

6

Difference of Two Squares

a² − b² = (a + b)(a − b). A special factorisation pattern.

7

Parabola

The U-shaped curve produced when graphing a quadratic function. Opens upward if a > 0, downward if a < 0.

8

Roots / Solutions

The values of x where the quadratic equals zero. Graphically, these are the x-intercepts of the parabola.

Active recall

Close the notes and answer these

  1. 1.Without looking, explain quadratic equation and give one example or consequence.
  2. 2.Without looking, explain factorising to solve and give one example or consequence.
  3. 3.Without looking, explain quadratic formula and give one example or consequence.
  4. 4.Without looking, explain discriminant and give one example or consequence.

Content reviewed 23 July 2026 against the current linked specification.

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