GCSE Maths Quadratic Equations Notes
A quadratic has an x² term. You can solve quadratics by factorising, by the quadratic formula, or by completing the square.
Key concepts
- Quadratic equation
- An equation of the form ax² + bx + c = 0 where a ≠ 0. It usually has two solutions (roots).
- Factorising
- Find two numbers that multiply to c and add to b, e.g. x² + 5x + 6 = (x + 2)(x + 3).
- The roots
- The x-values where the curve crosses the x-axis. Set each bracket equal to zero to find them.
Key formulas
Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a
Use when the quadratic will not factorise neatly.
Completing the square
x² + bx = (x + b/2)² − (b/2)²
Worked example
Solve x² + 5x + 6 = 0.
- 1Find two numbers that multiply to 6 and add to 5: 2 and 3.
- 2Factorise: (x + 2)(x + 3) = 0.
- 3Set each bracket to zero: x + 2 = 0 or x + 3 = 0.
- 4Solve each: x = −2 or x = −3.
Answer: x = −2 or x = −3
Common mistakes
✗ Forgetting the second solution.
✓ A quadratic normally has two roots — give both unless told otherwise.
✗ Sign errors in the factors, e.g. (x + 2)(x − 3) for x² + 5x + 6.
✓ Check by expanding: the numbers must multiply to +6 and add to +5, so both are positive.
✗ Using the quadratic formula but mishandling the −b or the ± .
✓ Write a, b and c down first, then substitute carefully; compute both the + and − versions.
Quick check
Try these, then reveal the answers.
Factorise x² + 7x + 12.Reveal answer
(x + 3)(x + 4)
Solve x² − 9 = 0.Reveal answer
x = 3 or x = −3
Solve (x − 4)(x + 1) = 0.Reveal answer
x = 4 or x = −1
Exam tips
- Try factorising first — it is fastest. Only reach for the formula if the numbers are awkward.
- If the question says 'give your answer to 2 decimal places', that is a strong hint to use the quadratic formula.
- Sketching the parabola helps you check your roots look sensible.
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Frequently asked questions
When should I use the quadratic formula instead of factorising?+
Use the formula when the quadratic does not factorise into whole numbers, often signalled by 'give your answer to 2 decimal places'.
Why does a quadratic have two answers?+
Its graph is a parabola that usually crosses the x-axis twice, so there are two x-values that satisfy the equation.
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