Friday, August 19, 2016 0

Whether you are taking HL, SL or Studies, you can be certain that there will be at least one quadratic expression to factorise in your exams – probably several. Here is my quick guide.

If I multiply out these brackets:

I will get:

The result is a quadratic expression: a quadratic must have a term in , and can also have a term in and a constant (a number). If you reverse the process, you will be factorising the quadratic. So, here are the quadratic expressions you might be faced with:

This is easy because all you have to do is find the common factor. For example:

Quadratic with no term in x

Generally these quadratics cannot be factorised, unless in the form of the difference of two squares, ie . This factorises to . For example:

It is important to remember that expressions of the form  do not factorise. For example, since

.

Remember that when factorising any expression you should look for a common factor first:

If the coefficient of x2 is 1:

Look at the expansion at the top: the 2x came from 3x – and -3 from 3. So in reverse it’a quite simple: the two numbers in the brackets will add to give the term in x and multiply to give the constant term. You just have to be really careful with negative numbers. So, some examples:

Example 1:   Factorise .

So, what two numbers add to give 10 and multiply to give 21? Answer: 3 and 7.

Example 2:   Factorise .

What two numbers add to give -2 and multiply to give -15? Answer: -5 and 3.

Example 3:   Factorise

Since the numbers multiply to give a positive, but add to give a negative, they must both be negative. They are therefore -5 and -1. (It would be very easy to come up, incorrectly, with -2 and -3 – it just shows how important it is that you check your answer).

If the coefficient of x2 is greater than 1:

There is a method for doing these, but it’s quite complicated to remember. The questions are rarely too challenging, so trial and error should get you there – but don’t forget you can always use the quadratic formula if you need to. Also, you must first check whether there is a common factor, which will simplify things considerably:

Example 4:   Factorise

The brackets will begin with 3x and x, so that means the number in the second bracket will be multiplied by 3. The two numbers are probably 5 and 2, with one of them being negative. Trying all the possibilities quickly leads to the correct answer:

Example 5:   Factorise

5 is a common factor:

So far we have been factorising quadratic expressions. Quadratic equations can be solved by factorisation, but only if the right hand  side is zero. This is because when , either a or b must be zero. So if:

then either (x – 2) = 0, giving x = 2, or (x + 5) = 0, giving x = -5.

Example 6:   Solve

First rearrange to make the right hand side equal zero:

Now factorise:

And solve:

or

Example 7:   Solve

There is no constant term, but there is a common factor:

or

I hope you found this useful. Do give me feedback if there are other topics you would like me to cover.

• August 20, 2016

Hello!

I’m a math studies student and these posts really help me out!

Would it be possible for you to write a post covering the “oh so confusing” topic of functions (more specifically exponential functions)?

Thank you once again for these informative posts. Keep up the good work!

• August 21, 2016

Hi Nathalie

Glad you find the posts about mathematics topics useful – I intend to do lots of these! There will definitely be one about exponential functions.

Ian