Factorising Quadratics When a ≠ 1

Factorising Quadratics

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 a ≠ 1.

If you’re sitting the higher paper for your GCSE, you need to be able to factorise quadratics in the form ax^2+bx+c. This is more difficult when a 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 revision sheet for exam-style questions on this topic.

The Partition Method

Let’s look at factorising 15x^2-x-6 .

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

Now, we find the value of ac :

ac = 15 × -6
     = -90

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

The Partition Method

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: 9x and -10x.

15x^2-x-6=15x^2+9x-10x-6

Next, we find the highest common factor of 15x^2 and 9x and factorise it out of the first two terms.

3x(5x + 3) - 10x - 6

Then, we do the same for the final two terms. It’s important the expression in the brackets is the same, (5x + 3) , so we take out a factor of -2.

3x(5x + 3) - 2(5x + 3)

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

15x^2-x-6=(3x-2)(5x+3)

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

The Grid Method

Let’s look at factorising the same quadratic: 15x^2-x-6 .

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

Now, we find the value of ac :

ac = 15 × -6
     = -90

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

The Grid Method

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 x^2 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 9x and -10x .

Factorising quadratics when a is not 1 - Grid Method explained pt. 1

We find the highest common factor of 15x^2 and 9x , which is 3x . We write this to the left of the top row:

Factorising quadratics when a is not 1 - Grid Method explained pt. 2

Then, we consider each term across the top and find the term we need to multiply 3x by to get this term. For example, 3x × 5x = 15x^2 , so we write 5x above 15x^2 .

Pt. 3

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

Pt. 4

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.

15x^2-x-6=(3x-2)(5x+3)

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. 6x^2+13x+5

2. 21x^2+29x-10

3. 36x^2-60x+25

1. (2x + 1)(3x + 5)

2. (7x - 2)(3x + 5)

3. (6x - 5)^2 or (6x - 5)(6x - 5)


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