Factorising Quadratics

So today we are looking at factorising quadratics. Now a quadratic is an algebraic expression that looks like this:

x² + 5x + 6

Now the first thing we have to remember is that factorising is the opposite of expanding. So when we were doing expanding and we had something like (x + 3)(x + 2), we would expand it using FOIL and get:

x² + 2x + 3x + 6

We would then collect like terms and get a final answer of:

x² + 5x + 6

This is a quadratic.

Since factorising is the opposite of expanding, we want to start with an expression like x² + 5x + 6 and manipulate it so we then get (x + 3)(x + 2). Basically all we want to find out is the numbers that go in the brackets. We know there will always be the x’s [i.e. (x + ….)(x + ….)], we need to find out the numbers that go after the x’s.

If we look at the example above, we can see that these numbers in the brackets are 2 and 3. If we look carefully we can see that the numbers 2 and 3 in our brackets add together to give 5, which is the number out the front of our second term from our expression. We also can see that 2 and 3 multiply together to give 6, which is our third (last) term in our expression. This tells us that the numbers in the brackets have to:

• ·         Add together to give the second term
• ·         Multiply together to give the third term.

So looking at an example:

Factorise x² + 11x + 24

First look at pair of numbers that multiply together to give our third term, 24 (i.e. factors of 24). These are:

·         1 and 24

·         2 and 12

·         3 and 8

·         4 and 6

Now we need to pick one of these pairs that add up to give our second term, 11. This can only be 3 and 8. We now know that the numbers in our brackets are 3 and 8. So our answer, fully factorised, is:

(x + 8)(x + 3)

So that is how we factorise quadratics. A key point to remember is that factors of our third term may also be negative. The following website shows examples of these: http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-factorisingquadratics.pdf -> (only look up to and including section 3). 

Remember if you do have any question on this, feel free to comment on this post or email me. There is a worksheet on this topic in the "Worksheets" tab.