Solving Quadratic Equations for GCSE Maths: Methods, Examples and Questions

Quadratic equations are a key topic in GCSE Maths, and understanding how to solve them is essential for exam success. This page provides clear explanations, worked examples, and practice questions to help you master solving quadratics.

You'll learn three main methods for solving quadratic equations: solving quadratics by factorising, solving quadratics using the quadratic formula, and solving quadratics by completing the square. Each method is explained step by step, with video tutorials to guide you through the process.

Whether you're revising for an exam or just need extra practice, this page has everything you need to build confidence in solving quadratic equations.

Solving Quadratic Equations by Factorising

Usually the easiest and fastest method to solve a quadratic is by factorising.

If we can write a quadratic as the product of two brackets, then we can easily solve it.

For example, if x² + 5x + 6 = 0, we can rewrite x² + 5x + 6 as (x + 2)(x + 3) and so (x + 2)(x + 3) = 0.

Since multiplying these two brackets results in 0, then either x + 2 = 0 or x + 3 = 0, giving us x = -2 and x = -3.

Watch this video to learn how to solve quadratics by factorising.

Solving a Quadratic Equation with the Quadratic Formula

Not all quadratics can be easily factorised so there are alternative methods in this case. One of them is the dreaded quadratic formula!

The quadratic formula causes fear into even the toughest students at first sight.

It's a long formula which includes many operations but luckily you don't need to memorise it!

By writing the equation in the form ax² + bx + c = 0, you just need to identify the values of a, b, and c and then substitute them into the formula.

You'll usually need to use a calculator to find the solutions as you'll rarely have integer answers.

One of the advantages of this method is that it makes it easy to solve quadratics where the coefficient of x² is not 1. Eg. 2x² + 5x + 1 = 0 or 3x² - 5x - 3 = 0.

In this video, you'll learn how to solve a quadratic using the quadratic formula.

Solving Quadratics by Completing the Square

'Completing the square' means to write a quadratic equation in the form (x + a)² + b.

Once you know how rewrite an equation in completed square form, there are usually just three quick steps to solve the equation.

Although less commonly used, many students prefer this method over the quadratic formula as it reduces the risk of calculator mistakes, especially with negative numbers.

In this video, you'll learn how to solve quadratics by completing the square.

The rest of the lesson on differentiation is not quite ready but if you already know what you're doing we have practice questions linked below.