How to I factorise a quadratic equation?

I think the best way to show how to do this is with an example

"Factorise 6x² + 7x - 3"

Looking at that expression, we can't factorise it straight away, so we'll have to rewrite it in a way that allows us to factorise it

We'll start by labelling the coefficients of x as a, b, and c

So a = 6  b = 7 and c = -3

Next, multiply a and c 

6 x -3 = -18

Now you need to find 2 numbers that multiply to give ac (which here is -18) and add to give you b (in this case, 7)

In this case, these two numbers are 9 and -2. We are going to use these two numbers to rewrite the original epression as:

6x² + 9x - 2x -3

Basically what we've done here is to rewrite 7x as 9x -2x becuase it allows us to split up the quadratic into 2 expressions which are much easier to factorise:

6x² + 9x = 3x (2x+3)

-2x -3 = -1(2x+3)

Once factorised, each expression will have an identical set of brackets. This forms the first half of the factorised expression. The second is formed by combining the 2 terms outside of the brackets.

(2x+3)(3x-1)

What happens if one expression won't factorise?

There is a nice easy solution to this, BUT you can only use it when factorising quadratics. You can simply take out a factor of 1. In the example here, we took out a factor of -1 and the contents of the bracket were (3x-1), If you take out a factor of 1, the contents become (ax+1)

What happens if I can't factorise either equation?

Try swapping the two numbers you rewrote b as. For example, instead of writing 9x - 2x, try writing -2x + 9x (In this case they both work, but they won't always)

What happens if the 2 brackets aren't the same?

Unfortunately, this means you've made a mistake somewhere in your earlier calculations, so you'll need to go back and check those. The most common one is missing out (or even adding in) a negative when writing a, b and c or when multiplying a and c.