How To Easily Factorise A Quadratic Equation

Factorising quadratic equations is a skill that is tested as soon as KS3 and is necessary all the way through KS5, and is prominent in GCSE exams and A - Levels. In some cases, it`s clear that you`re meant to factorise, so let`s go through an example like this, and then find a challenging GCSE past paper question.

Question a) x² + 7x + 13 = 3

Step 1) With any quadratic equation, the first step is to set 1 side equal to 0. This can be done by subtracting 3 from both sides.

x² + 7x + 10 = 0

Step 2) We want to rewrite the left hand side in the form of (Ax + C) (Bx + D). In order to do this, let`s first rewrite our quadratic in the form of ax² + bx + c = 0 (Note that when factorised, the letters are uppercase and in the quadratic, they are lowercase. This means they are not the same numbers)

x² + 7x + 10 = 0

ax² + bx + c = 0

a= 1, b=7, c=10

Step 3) Let`s begin factorising.

We know we will end up with something in the form of (Ax+C) (Bx+D). Using our standard rules for expanding brackets, the x² term will have a coefficient of A x B. Looking at our quadratic equation, the x² term has a coefficient of 1. Therefore, A x B must = 1, so both A and B are 1.

We can write our factorised brackets as (1x + C)(1x+D), or simply (x+C)(x+D). If we multiplied this out, we would get x² + Cx + Dx + CD. Therefore, the coefficient of the x term = c + d. Additionally, the last term in the quadratic (the term with no x next to it) = CD

In other words, b = C + D, and c= C x D

As we wrote earlier, b = 7 and c = 10.

The easiest way to find C and D are to make a list of pairs of numbers that multiply to make 10 (c), and add to make 7 (b).

The pairs of numbers that multiply to 10 are 1 and 10, 2 and 5. As 2 and 5 add to 7, these are the correct numbers for C and D.

Therefore, x² + 7x + 10 = 0 can be written as

(x+2)(x+5) = 0

This method can work NO MATTER the numbers in front of the x² term, and no matter what the signs are. Lets do a harder example with the same method.

GCSE 2022 PAPER 2 QUESTION 23

Factorise 3x² - 16x -12 = 0

Step 1) The quadratic is already equal to 0

Step 2)

3x² - 16x -12 = 0

a=3, b=-16, c=-12

Step 3) Let`s factorise. Note, this step is slightly different as the brackets will have a minus sign.

(Ax+ C)(Bx - D) = ABx² -ADx + BCx - CD

= 3x² -16x - 12

Therefore, A x B = 3. So we know A = 3 and B = 1 (For now, it doesn`t matter which way round)

(3x+C) (1x-D) = 3x² - 3Dx + Cx - DC

= 3x² -16x -12

Therefore, C - 3D = -16 and D x C =12 ( If -DC = -12, DC = 12)

The list of numbers that multiply to 12 are (1 x 12) (2 x 6) and (3 x 4)

If we look at the ones where multiplying one of them by -3 and adding the other = -16, we have our answer. Again, this is trial and error. IF NONE OF THEM WORK OUT, IT MEANS WE PUT A AND B THE WRONG WAY ROUND AND NEED TO REPEAT THIS STEP

In this case, we did it the right way round, as (6 x -3) + 2 = -18 + 2 = -16

Therefore, c = 2 and D = 6

3x² - 16x -12 = (3x + 2) (x - 6)

Well done!! You can now factorise a quadratic equation! Good luck, keep practicing, and message me if you need any help!!