Here is a simple guide to factorising quadratics on the GCSE mathematics foundation exam.

If you’re doing the maths foundation exam – luckily for you, you’re only going to need to factorise quadratics of the form x^{2} + ax + b (which trust me, makes things a LOT easier! If you’re looking for a guide to factorising quadratics of where there is a number before the x^{2}, check back for a similar blog in the coming weeks).

This blog will be focused on quadratics of the form x^{2} + ax + b, where b is NOT zero (if b is zero, the quadratic will actually factorise into just one bracket). This means we know the answer will always be in the form (x )(x ), where the numbers inside the brackets are to be determined.

Let’s walk through an example:

1. Factorise x^{2} + 7x + 6.

We know the answer is (x )(x ), we just have to figure out what the two numbers inside the brackets are.

The two numbers will **multiply to get the end number** (6), and **add to make the middle number** (7).

If you don’t know it straight away, then the best course of action is to list all pairs of numbers that multiply to get 6, and then work out which of them add to get 7.

These pairs multiply to get 6:

2 & 3 2 + 3 = 5

1 & 6 1 + 6 = 7

(If you’re sharp you may notice that there are actually two more pairs, -2 & -3, and -1 & -6. However, because the quadratic has no negative numbers in it, we can ignore these two).

So, 1 and 6 do the job.

Our final answer is (x + 1)(x + 6).

You can check your answer here – expand the double bracket and you SHOULD get x^{2} + 7x + 6. If you don’t, then something has gone wrong somewhere.

(x + 1)(x + 6) = x^{2} + x + 6x + 6

= x^{2} + 7x + 6

This matches the quadratic we were asked to factorise, so we know we have the correct answer.

Let’s do another…

2. Factorise x^{2} + 8x + 12

We are looking for two numbers that **multiply to get 12** and **add to get 8**.

These pairs multiply to get 12:

1 & 12 1 + 12 = 13

2 & 6 2 + 6 = 8

3 & 4 3 + 4 = 7

So, it’s going to be 2 & 6.

Our answer is (x + 2)(x + 6).

Let’s expand to check…

(x + 2)(x + 6) = x^{2} + 2x + 6x + 12

= x^{2} + 8x + 6

This matches, so all’s good!

3. Factorise x^{2} + 9x + 20

We are looking for two numbers that **multiply to get 20** and **add to get 9**.

These pairs multiply to get 20:

1 & 20 1 + 20 = 21

2 & 10 2 + 10 = 12

4 & 5 4 + 5 = 9

So it’s going to be 4 & 5.

Our answer is (x + 4)(x + 5).

Check it…

(x + 4)(x + 5) = x^{2} + 4x + 5x + 20

= x^{2} + 9x + 20

Nailed it!

Okay, so hopefully that made sense?

As always with maths, something may seem easy, but there is always a more difficult type of question. And invariably this occurs when we introduce our old friend… the minus sign! Take this example…

4. Factorise x^{2} + 2x – 8.

Now, we follow the same process as before, but the reason this is going to be harder is because there will be more combinations.

We are looking for two numbers that **multiply to get -8** (don’t forget about the minus) and **add to get 2**.

These pairs multiply to get -8:

-1 & 8 -1 + 8 = 7

-8 & 1 -8 + 1 = -7

-2 & 4 -2 + 4 = 2

-4 & 2 -4 + 2 = -2

(Notice that for each pair of factors, there are two combinations – with the positive and negative signs switched)

So it’s going to be -2 & 4.

Our answer is (x – 2)(x + 4).

Let’s check it…

(x – 2)(x + 4) = x^{2} – 2x + 4x – 8

= x^{2} + 2x – 8

Last one (I promise…):

5. Factorise x^{2} – x – 12.

We are looking for two numbers that **multiply to get -12** and **add to get -1** (-x means -1x, but mathematicians are lazy so don’t bother writing the 1!).

These pairs multiply to get -12:

-1 & 12 -1 + 12 = 11

-12 & 1 -12 + 1 = -11

-2 & 6 -2 + 6 = 4

-6 & 2 -6 + 2 = -4

-3 & 4 -3 + 4 = 1

-4 & 3 -4 + 3 = -1

So it’s going to be -4 & 3.

Our answer is (x – 4)(x + 3).

Let’s check it…

(x – 4)(x + 3) = x^{2} – 4x + 3x – 12

= x^{2} – x – 12

So there it is. It’s quite easy really, just needs a bit of practice.

If you want to work through more examples with a qualified maths tutor to help your understanding, you can book a free taster session here.

For more algebra problems, try our algebra worksheet.

Work through these, post your answers in the comments below or email them to sam@metatutor.co.uk and I’ll let you know if you’re right – look out for question 10!

Factorise the following quadratics:

- 1. x
^{2}+ 7x + 10 - 2. x
^{2}+ 8x + 15 - 3. x
^{2}+ x – 6 - 4. x
^{2}– 7x – 8 - 5. x
^{2}+ 10x + 24 - 6. x
^{2}– x – 20 - 7. x
^{2}– 3x – 18 - 8. x
^{2}+ 11x + 28 - 9. x
^{2}– 4x – 12 - 10. x
^{2}– 5x + 6

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