How to Expand And Factorise Quadratics
Introduction
For GCSE Maths and this applies to the higher and foundation paper you need to be able to know how to expand and factorise quadratics. Whenever you see a set of two brackets in GCSE Maths then the contents in each bracket need to be multiplied together.
In order to factorise a quadratic you will:
- Have a set of two brackets, or
- Have taken out a common factor
When it comes to finding a common factor, this can be a number, a letter(s) or a combination of the two.
You should also be able to spot a quadratic which is in the form of a difference of two squares i.e. a^2-b^2=(a-b)(a+b)
It is also important that you are comfortable with collecting like terms.
How to Expand And Factorise Quadratics - Example
Take a look at the following question:
This is a quadratic but there are only two terms. Here you need to identify common factors.
First look at the numbers. What is common to 2 and 4? Well 2 is common to both.
Next look at the letters. What is common to x^2 and x y?
Well is x is common to both.
So the common factors are 2x.
This can be written as: 2 x(?-?)
The question to ask now is what goes inside the brackets?
There will be two terms inside the bracket, because the expression has two terms.
The first term of the original expression is 2 x^2 because 2x is the factor, what can be multiplied with the factor to give 2 x^2?
The answer is x so this will be the first term inside the bracket.
The second term of the original expression is 4xy, because 2x is the factor, what can be multiplied with the factor to give 4xy? The answer is 2y so this will be the second term inside the bracket.
So the final factorised answer will be: 2xx-2y.
To be confident that your answer is correct you should always expand the brackets that you have obtained to see if you get the original expression.
How To Expand And Factorise Quadratics - Example
Example
Take a look at the following question:
When it comes to brackets, whatever is inside the bracket must be multiplied with whatever is outside the bracket.
For the first bracket: 32a+5=6a+15. Note that there are two terms inside the brackets so there should be two terms after the multiplication also.
For the second bracket: 5a-2=5a-10
So after the expansion you will have: 6a+15+ 5a-10.
Even though the brackets have been expanded, the question is not yet complete. In order to fully simplify the expansion, like terms must now be collected to give: 11a-5, which is the final answer.
How to Expand And Factorise Quadratics – Example
Take a look at the following question:
This is an example of a difference of two squares which you must be able to recognise.
It is important that you are able to recognise square numbers because 49=7^2
\therefore x^2-49=x^2-7^2=(x-7)(x+7)Example
Take a look at the following question:
In order to factorise a quadratic you generally need to compare it to the general quadratic equation which is a x^2+b x+c.
(It does not matter that the letters are not the same).
In this case a=1, b=-10, c=16
You need to find two numbers which add up to -10 and the same two numbers when multiplied together give 16.
Well -8+-2=-10,-8 \times-2=16
So the quadratic can be written as: y^2-8 y-2 y+16.
There are now 4 terms but the quadratic is still the same.
y^2-8 y-2 y+16Factorising each half the quadratic will give: y(y-8)-2(y-8)
You will see that y-8 is now common so the fully factorised answer is:
(y-8)(y-2)Question Practice
Try the following question on your own before looking at the solution.
Question Practice Solution
a) Here 5 is the common factor, \therefore 5(x-2)
b) Here the common factor is 2 p, \therefore 2 p(p-2 q)
c) Here the brackets needs to be multiplied together to give: t^2-4 t+5 t-20=t^2+t-20. Remember to collect like terms once the expansion has been done.
d) Here you need to find two numbers that when added together gives 17 and when multiplied gives 60. The numbers are 12 and 5. The factorised form will then be (x+12)(x+5). Remember you can always expand the brackets obtained to check that you arrive back at the expression that you started with.
e) This you need to recognise as the form of a difference of two square where 144=12^2 \cdot x^2-12^2=(x-12)(x+12)
Being able to factorise expressions is a very important skill and if you are considering pursuing maths afterwards then even more so. You have seen a number of examples in this article and it is important that if you do not understand any of the questions or techniques then you have another go.
Factorising of algebraic expressions in GCSE Maths is something that always appears and it does not need to be a difficult topic if you follow some of the suggestions in this article.