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Solving Quadratic Equations: Factorising and the Formula

Solve quadratics by factorising and with the quadratic formula, using worked examples. Covers GCSE quadratic equations and Algebra 1 and 2 methods.

  • ⏱ 3 min read
  • ✏️ 3 worked examples
  • 🎯 5 practice questions
  • 📶 Intermediate
GCSEA-LevelHigh School
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Curriculum fit: GCSE Maths Algebra (Foundation: factorising; Higher: formula and completing the square). US: Algebra I and II quadratic equations.

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What Is It?

A quadratic equation can be written as ax² + bx + c = 0. It can have two solutions, one solution or none, and you can solve it by factorising, by completing the square or by using the quadratic formula.

Always start by rearranging so that one side equals zero. Factorising is quickest when it works, and the formula works every time.

x² + 5x + 6 = 0(x + 2)(x + 3) = 0x = −2 or x = −32 × 3 = 6 and 2 + 3 = 5, so the brackets are (x + 2)(x + 3)Discriminant b² − 4acpositive2 solutionszero1 solutionnegativeno real solutionsx = (−b ± √(b² − 4ac)) ÷ 2a
Factorise when you can: find two numbers that multiply to c and add to b. Otherwise use the quadratic formula, and check the discriminant first.

Key Formulas and Ideas

  • Factorising: find two numbers that multiply to give c and add to give b (when a = 1).
  • If (x − p)(x − q) = 0, then x = p or x = q.
  • Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a.
  • Discriminant b² − 4ac: positive means two solutions, zero means one, negative means none.

Worked Examples

Example 1

Factorising

Question

Solve x² + 7x + 12 = 0.

  1. Find two numbers that multiply to 12 and add to 7: 3 and 4.
  2. (x + 3)(x + 4) = 0.
  3. x = −3 or x = −4.

✓ Answer x = −3 or x = −4

Example 2

Difference of Two Squares

Question

Solve x² − 9 = 0.

  1. Factorise: (x − 3)(x + 3) = 0.
  2. x = 3 or x = −3.

✓ Answer x = 3 or x = −3

Example 3

Quadratic Formula

Question

Solve 2x² − 5x − 3 = 0.

  1. a = 2, b = −5, c = −3.
  2. Discriminant: 25 + 24 = 49, and √49 = 7.
  3. x = (5 ± 7) ÷ 4, giving x = 3 or x = −1/2.

✓ Answer x = 3 or x = −1/2

Common Mistakes to Avoid

  • Not rearranging to equal zero before solving.
  • Forgetting the ± in the formula, which gives only one solution.
  • Mixing up signs when substituting b into the formula.
  • Giving only one solution when there are two.

Practice Questions

0 of 5 answered
Q1Solve x² + 3x + 2 = 0.
Q2Solve x² − 16 = 0.
Q3Solve (x − 1)(x + 4) = 0.
Q4What is the discriminant of x² + 2x + 5?
Q5Solve x² − x − 6 = 0.

Frequently Asked Questions

What does the discriminant tell me?

If it is positive there are two real solutions, if it is zero there is one, and if it is negative there are no real solutions.

How do I decide which method to use?

Try factorising first. If it does not factorise easily, use the formula or complete the square.

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Sources and Review

Reviewed by StudyMorrow editorial review on 5 Nov 2026.