Exam-takers, note-makers and revision-breakers, there comes a time in GCSE Higher Level Maths when you need to demonstrate a competent understanding of factorising quadratics when ≠ 1.

If you’re sitting the higher paper for your GCSE, you need to be able to factorise quadratics in the form . This is more difficult when is not 1.

In this blog, we will talk about two methods to factorise these expressions – the partition method and the grid method. We’ve included some practice questions too!

Check out our Expanding and Factorising bundle for exam-style questions on this topic an many more.

Higher Expanding and Factorising Digital Revision Bundle

The Partition Method

Let’s look at factorising .

The first step is to identify the values of , and . In this case, = 15, = -1 and = -6. Make sure you include any negative signs in these values.

Now, we find the value of :

= 15 × -6
     = -90

We are looking for a pair of numbers that multiply together to give (-90) and add together to give (-1). The easiest way to find the numbers is to write out the factors of -90:

The only pair that adds together to give -1 is 9 and -10.

We use these two numbers to split the middle term into two terms: 9 and -10.

Next, we find the highest common factor of and and factorise it out of the first two terms.

Then, we do the same for the final two terms. It’s important the expression in the brackets is the same, , so we take out a factor of -2.

Since the expressions in the two brackets are the same, we can combine the terms in front of them to make our second bracket:

This is our final answer! You can check it by expanding the brackets.

The Grid Method

Let’s look at factorising the same quadratic: .

Just as with the partition method, the first step is to identify the values of , and . In this case, = 15, = -1 and = -6. Make sure you include any negative signs in these values.

Now, we find the value of :

= 15 × -6
     = -90

We are looking for a pair of numbers that multiply together to give (-90) and add together to give (-1). The easiest way to find the numbers is to write out the factors of -90:

The only pair that adds together to give -1 is 9 and -10.

This is where the methods are different. We write the expression in a grid with the term in the top left corner and the constant term in the bottom right corner. We then use our two numbers, 9 and -10, to fill the remaining two boxes with 9 and -10.

We find the highest common factor of 15 and 9, which is 3. We write this to the left of the top row:

Then, we consider each term across the top and find the term we need to multiply 3 by to get this term. For example, 3 × 5 = 15, so we write 5 above 15.

Then, we do the same for the bottom row. We simply find what we multiply 5 by to get -10:

Our factorised expression is made up of 2 brackets – one from the terms across the top and the other from the terms to the side.

Factorising quadratics when a ≠ 1 can take some time to master. Try out these tasks to reinforce your learning.

Factorising Quadratics – Practice Questions

Have a go at these questions to practise your preferred method.

1.

2.

3.

Show Answers

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