How To Do GCSE Maths Algebraic Proof Fast
Introduction
Questions involving GCSE Maths algebraic proof must be done carefully with all steps being shown.
You need to be aware of the different types of wording that is used in proof questions.
If you see the word “verify” then you simply need to perform a substitution in order to show that the expression works.
If you see the word “show” then you need to show that both sides of an expression are algebraically correct.
If you see the word “prove” then you have to see that one side (either the left or the right) is the same as the other side.
Example
Take a look at the following GCSE Maths Algebraic proof question:
Given that n^2+(n+1)^2-(n+2)^2=(n-3)(n+1)
a) Verify that the expression is correct
b) Show that the result is true
c) Prove that the result is true
Solution
a) In order to verify that the expression is correct you can simply substitute any number into both sides of the expression. Taking n = 1 this will give:
\begin{aligned} & 1^2+(1+1)^2-(1+2)^2=(1-3)(1+1) \\ & 1+4-9=(-2)(2) \\ & -4=-4 \end{aligned}This is enough to verify that the expression is correct.
b) Expanding both sides of the expression gives:
\begin{aligned} & n^2+n^2+2 n+1-\left(n^2+4 n+4\right)=n^2+n-3 n-3 \\ & n^2-2 n-3=n^2-2 n-3 \end{aligned}This shows that both sides of the expression are the same.
c) Expanding the left hand side gives:
n^2+n^2+2 n+1-\left(n^2+4 n+4\right)=n^2-2 n-3=(n-3)(n+1)=\text { RHS }This has proved that the expression is correct.
GCSE Maths Algebraic Proof Example
Take a look at the following GCSE Maths algebraic proof question:
There are two ways that you could approach this question.
If the nth even number is 2n then this means that the next number will be 2n + 1, which will be odd. So 2n + 2 must be the next even number.
Alternatively you can go back to the definition of an even number which is a number that is a multiple of 2.
2n is a multiple of 2 🡪 so this is even.
2n + 1 cannot be factorised and so cannot be a multiple of 2 🡪 so this is not even.
2n + 2 = 2(n + 1) is a multiple of 2 🡪 so this is even.
Example
Take a look at the following question:
Going back to the definition of an even number, a number that is a multiple of 2, then three consecutive even numbers will be 2n, 2n +2 and 2n +4.
Adding these together will give:
2n + 2n + 2 + 2n + 4 = 6n + 6 = 6(n + 1)
Because this has been factorised with 6 as the common term, this is telling you that 6(n +1) is indeed a multiple of 6.
Example
Take a look at the following question:
Here you need to expand the brackets and to collect like terms as follows:
\left(9 n^2+6 n+1\right)-\left(9 n^2-6 n+1\right)=9 n^2+6 n+1-9 n^2+6 n-1=12 n=4(3 n)This shows that the result is a multiple of 4 and so the proof has been successful.
Question Practice
Try the following questions on your own before looking at the solution:
Question Practice Solution
Let the first whole number be n so the next consecutive whole number will be n + 1.
Adding these together will give:
n + n + 1 = 2n + 1
Now 2n is a number that is a multiple of 2 so it is even.
If you add 1 to an even number that it is always odd.
So 2n + 1 proves that the sum of two consecutive whole numbers will always be odd for all values of n.
Question Practice
Try the following questions on your own before looking at the solution:
Question Practice Solution
First you need to expand the brackets and to collect any like terms:
\begin{aligned} & \left(25 n^2+10 n+1\right)-\left(25 n^2-10 n+1\right)=25 n^2+10 n+1-25 n^2+10 n-1=20 n= \\ & 5(4 n) . \end{aligned}Here the result shows that the outcome is a multiple of 5.
Question Practice
Try the following questions on your own before looking at the solution:
If 2n is an even number then the next even number will be 2n + 2.
Squaring these will give: (2 n)^2+(2 n+2)^2=4 n^2+4 n^2+8 n+4=8 n^2+8 n+4=4\left(2 n^2+2 n+1\right), which shows that the result is a multiple of 4 and so the proof has been successful.
With most cases of proof you are using algebra to help you obtain the desired result. And because you are using algebra you need to be careful with your expansion and simplifying of terms.
GCSE Maths Algebraic Proof can be quite interesting as it allows you to show that specific results are true such as showing that a certain expression is a multiple of 4.