How To Solve Algebraic Fractions Quickly
Introduction
Algebraic fractions follow the same rules of arithmetic fractions but the following are important techniques that you must remember:
Being able to factorise a quadratic equation where x^2 has a unit value
Being able to factorise a quadratic equation where x^2 does not have a unit value
Being able to recognise a difference of two squares
Being able to apply the rules of arithmetic fractions to algebraic fractions
GCSE Maths Algebraic Fractions: Example
Take a look at the following question:
\text { Simplify fully } \frac{x^2+x-6}{x^2-7 x+10}Here you will see that the numerator and denominator are both quadratic equations and they should both be factorised.
\begin{gathered} x^2+x-6=(x+3)(x-2) \\ x^2-7 x+10=(x-5)(x-2) \end{gathered}This means that the algebraic fraction can be written as follows:
\frac{(x+3)(x-2)}{(x-5)(x-2)}The next part is to see what can be cancelled. It is important that you must only cancel terms that are a product of something. The factorised forms of the quadratics show brackets and brackets indicate terms that are multiplied together.
Now (x-2) appears in both the numerator and the denominator and so this can be cancelled to leave: \frac{(x+3)}{(x-5)}
A common mistake when working with algebraic fractions is to now cancel out the x terms. This cannot be done because there are no terms that are being multiplied and if you were to substitute some values of x you will see that this does not work.
So the final answer is \frac{(x+3)}{(x-5)}
Another GCSE Maths Algebraic Fractions Example
You can see that the numerator and the denominator contain quadratic equations but they both only contain two terms. It is because they contain two terms that cause a lot of confusion. These can still be factorised by looking for common factors.
\begin{gathered} 6 x^2+3 x=3 x(2 x+1) \\ 4 x^2-1=(2 x)^2-1=(2 x-1)(2 x+1) \end{gathered}The denominator is a difference of two squares and this is something that you must be able to spot as there are no common factors for 4 x^2-1 other than 1.
The algebraic fraction can then be written as:
\frac{3 x(2 x+1)}{(2 x-1)(2 x+1)}In order to simplify fully, you can see that (2 x+1)
appears within the numerator and denominator and so can be cancelled.
The final answer is then: \frac{3 x}{(2 x-1)}
Question Practice
\text { Simplify fully } \frac{x+3}{4}+\frac{x-5}{3}In this question you need to apply the rules for the addition of arithmetic fractions.
The first thing that should be done is that a common denominator needs to be determined which is 12.
\frac{3(x+3)+4(x-5)}{12}=\frac{3 x+9+4 x-20}{12}=\frac{7 x-11}{12}And this is the final answer.
Question Practice
Try the following question on your own before looking at the solution.
\text { Simplify fully } \frac{2 x^2+3 x+1}{x^2-3 x-4}Question Practice Solution
So how did you get on? Hopefully you found that the answer to be \frac{2 x+1}{x-4}
Now what makes this tricky is the fact that the quadratic in the numerator is not a unit, yet you still need to be able to factorise quadratics such as these.
To factorise 2 x^2+3 x+1
you need to find two numbers that add up to 3 and which multiply together to give 2. These two numbers are 1 and 2.
2 x^2+3 x+1 can then be written as 2 x^2+x+2 x+1.
Here there are 4 terms but the quadratic is still the same.
Factorising each half gives:
x(2 x+1)+1(2 x+1)-(x+1)(2 x+1)Factorising the denominator: x^2-3 x-4=(x-4)(x+1)
So the algebraic equation can be written as follows: \frac{(x+1)(2 x+1)}{(x-4)(x+1)}
You will see that (x+1) appears in both the numerator and denominator and these can be cancelled to leave: \frac{(2 x+1)}{(x+1)}
And this is the final answer. No other terms can be cancelled because there is nothing that is a product of something else.
Factorising and simplifying algebraic fractions at GCSE Maths level can be tricky and you need to remember the process for actually factorising quadratics. It is especially more tricky when the first term of the quadratic is not 1 and this requires more mathematical skill.