So, you’ve discovered the joy of quadratics – or not – and you’ve realised they are a popular topic in the GCSE Maths Exam.
In one of my very first posts, I stated how critical times tables are to improve your maths.
If you want to be good at factorising you will need to be super speedy with your times tables, at least up to the 12 times table.
The more you know, the easier factorising becomes.
There are two types of quadratics you will need to factorise.
The first type are the ones that start with x2 i.e. there is always only 1 x2.
The second type is where you can have any number before the x2, and this gets tricky.
That number is called the coefficient of x2.
Let’s start with the the first type.
Here’s an example:
x2 + 5x + 6.
Here’s the way I do it:
Draw your brackets, with an x in each – that sorts out the x2.
(x )(x )
Now, the second and third part of the equation are both positive, so everything in the brackets should be positive.
(x + )(x + )
To choose the correct numbers that will complete the bracket, we need two factors of 6, that add up to make 5.
Easy – 3 x 2 =6, and 3+2 = 5
(x + 3)(x + 2)
If you multiply that out (always check your answers, especially in the exam where stress makes you do crazy things)
x2 + 3x + 2x + 6 which gives our original equation: x2 + 5x + 6
To summarise the method:
- Write out the xs and work out if you need negative or positive signs in the brackets
- List the factors of the last part of the equation and find the ones that add up to make the 2nd part
- Finish the brackets and check your answers
So why did we not use the factors 6 and 1?
To get 5x from this, we would have to subtract 1 from 6.
i.e. the brackets would need to look like this: (x + 6)(x – 1)
If we multiply that out we get: x2 + 6x – x -6 which gives x2 + 5x – 6.
Which is very similar to the original equation, but the last 6 needs to be positive not negative, that’s why in this case, using 6 and 1 in the brackets does not work.
The best way to get good at these is to practise, and practise consistently. Literally 10 mins per day spent on factorising quadratics for about 2 weeks, will make you an expert at these.
Let’s look at the second type:
These are where the coefficient of x2 can be any number.
For example: (stolen from the CGP GCSE Maths AQA textbook)
6x2 – 11x – 10
To do these you will need to add a few extra steps.
There are 2 ways you can get 6x2.
6x multiplied by x
or 3x multiplied by 2x
So our brackets could like this:
(x )(6x )
or (3x )(2x )
Then we need to choose the factors of the last part, in this case the factors of 10 and try them out in the brackets to see which gives us the original equation.
The process is similar to the first type of quadratic, but there is a bit more trial and error and problem solving involved.
However, as you do more of these questions, they do get easier and you will start to be able to find the numbers faster, without having to list every possibility.