Quadratic Equations

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What You Need to Know

The Big Picture

A quadratic equation is any equation that can be written as ax² + bx + c = 0 (where a ≠ 0). You'll learn three different methods to solve these equations, and you need to know when to use each one. Each method works for all quadratics, but some are faster or more appropriate depending on the question.

Why three methods?

Factorisation is quick when it works. The quadratic formula always works but needs careful handling. Completing the square unlocks deeper understanding and finds turning points. In exams, the question hints which method to use—always read carefully.

Method 1: Solving by Factorisation

The Zero Product Rule

If two things multiply together to give zero, then at least one of them must be zero. This is the heart of factorisation. Once you've written your quadratic as two brackets multiplied together, each bracket can equal zero, giving you two solutions (or sometimes one repeated solution).

Step-by-Step Process

1 Rearrange to standard form: ax² + bx + c = 0 (with zero on one side)

2 Factorise the left-hand side into two brackets

3 Set each bracket equal to zero separately

4 Solve each resulting equation to find both solutions

Example 1: Simple Factorisation

Solve: x² + 3x − 10 = 0

Find two numbers that multiply to −10 and add to 3. Those are 5 and −2. Factorise: (x − 2)(x + 5) = 0 First bracket: x − 2 = 0 → x = 2 Second bracket: x + 5 = 0 → x = −5 Solutions: x = 2 or x = −5

Example 2: Coefficients in the Brackets

Solve: 5x² − x = 0

Take out the common factor x: x(5x − 1) = 0 First bracket: x = 0 Second bracket: 5x − 1 = 0 → 5x = 1 → x = 1/5 Solutions: x = 0 or x = 1/5

When to use factorisation:

Use this method when the question explicitly asks you to "factorise and solve", or when you can spot the factors quickly. It's also essential for two-term quadratics (no c term) and difference-of-squares patterns like x² − 9 = 0.

Method 2: The Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a

This formula gives both solutions to any quadratic equation in the form ax² + bx + c = 0. Simply read off a, b, and c, substitute into the formula, and calculate.

How to Use the Formula

Step-by-Step Process

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Also in the full note

Method 3: Completing the Square
Deciding Which Method to Use
Summary & Review
Concepts Checklist
What to Memorize
Exam Tips & Common Mistakes
Test Yourself: Quick Practice
Example: Using the Quadratic Formula

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Quadratic Equations