This guide will tell you everything you need to know about factorising quadratic equations. Firstly, it’s important to recognise a quadratic equation. All quadratic equations can be written in the form: 

Quadratic equation

Where a, b and c are numbers. Number ‘a’ cannot be equal to 0, but ‘b’ and ‘c’ could be 0. 

There are several methods to solve quadratic equations, one of which is factorising. You need to know the method for factorising quadratics for both the foundation and higher tier papers. Let’s start with an example: 

Quadratic equation example 1

To factorise this equation we put it into double brackets, like this: 

Double brackets

To do this, there are a two main steps to follow… 

Factorising quadratic equations step one: 

Start by setting out the pair of double brackets. Then we will fill in the gaps. Where the coefficient of x squared is 1 (the number in the ‘a’ position), you should write x in the first position in each bracket:  

Now look at the last number of the original equation – the one in the ‘c’ position. In this case it’s a 5. Think of the factors of 5 (and list these in pairs to help your working). For 5 we only have one pair of factors: 1 and 5. You need to be able to add or subtract these numbers to make the coefficient of the x term (the middle number in the ‘b’ position); so here, 5 – 1 = 4. This gives us: 

Factorising quadratic equations step two: 

Then, to fill in the symbols, you need to look at how you add or subtract to make 4. In our 5 – 1 sum to make 4, the 5 is positive and the 1 is negative. Therefore: 

Solving the quadratic equation:

Once you have factorised the quadratic equation we have two expressions in brackets multiplying together to give 0. This means one of the brackets must be 0 (as anything multiplied by 0 will equal 0), so either: 

Solutions for quadratic equations

In the first case x would be 1 and in the second case x would be -5. With quadratic equations you will always have two possible answers for x. You write your answers like this: 

Solving quadratic equation

Full worked example

Here is another example with all of the working together: 

Quadratic equation example 2

What I have done here is look at the number 14 and thought of the pairs of factors of 14. The options are 1 and 14, or 2 and 7. 2 and 7 will add together to give me 9 (which is the coefficient of the x). So, 2 and 7 need to go into the brackets and both should be positive to give me a positive 9 when added and a positive 14 when multiplied. 

Check out our factorising quadratics worksheet for a range of questions you can practice with. You can also contact us directly for help with any maths topic. 

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