Solving Quadratic Equations

A quadratic equation contains an x^2 term as the highest power and has the general form ax^2 + bx + c = 0. Unlike linear equations, quadratic equations can have two solutions (roots). This topic is tested heavily on both Foundation and Higher tier AQA GCSE Mathematics papers. Foundation students need to solve by factorising, while Higher students must also use the quadratic formula and complete the square.

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Solving by Factorising

This is the most common method and is required at all levels. To solve a quadratic by factorising:

1. Rearrange the equation so that one side equals zero.
2. Factorise the quadratic expression.
3. Set each bracket equal to zero and solve.

Worked Example 1

Solve: x^2 + 5x + 6 = 0

Step 1: Factorise. Find two numbers that multiply to 6 and add to 5: 2 and 3.

x^2 + 5x + 6 = (x + 2)(x + 3) = 0

Step 2: Set each bracket equal to zero:

x + 2 = 0, so x = -2

x + 3 = 0, so x = -3

Answer: x = -2 or x = -3

Worked Example 2