One to one maths interventions built for GCSE success
Weekly online one to one GCSE maths revision lessons available in the spring term
In order to access this I need to be confident with:
Expanding brackets
Factors and multiples
Terms, Expressions, Equations, Formulas
Terms, Expressions, Equations, Formulas
Multiplying Negative Numbers
This topic is relevant for:
Difference of Two Squares
factorising quadratics in the form a2 – b2
We have previously learnt about factorising expressions into a single bracket and factorising quadratics into double brackets.
Here we will learn how to factorise a quadratic in the form
(Remember:
There are also factorising worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
What is the difference of two squares?
When we have a squared term subtracted from another squared term we can use the difference of two squares method. Square numbers are sometimes called perfect squares.
To use quadratic factorisation on an expression in the form
a2 – b2 = (a + b)(a – b)
How to factorise quadratics: difference of two squares
In order to factorise an algebraic expression using the difference of two squares:
- Write down two brackets
- Square root the first term and write it on the left hand side of both brackets
- Square root the last term and write it on the right hand side of both brackets
- Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.)
Factorising examples: difference of two squares
Example 1
Fully factorise
- Write down two brackets
2Square root the first term and write it on the left hand side of both brackets
3Square root the last term and write it on the right hand side of both brackets.
4Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter).
We can check the final answer by multiplying out the brackets!
Example 2
Fully factorise
Write down two brackets
Square root the first term and write it on the left hand side of both brackets
Square root the last term and write it on the right hand side of both brackets
Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.)
We can check the final answer by multiplying out the brackets!
Example 3
Fully factorise
Write down two brackets
Square root the first term and write it on the left hand side of both brackets
Square root the last term and write it on the right hand side of both brackets
Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.
We can check the final answer by multiplying out the brackets!
Example 4
Fully factorise
Write down two brackets
Square root the first term and write it on the left hand side of both brackets
Square root the last term and write it on the right hand side of both brackets
Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.)
We can check the final answer by multiplying out the brackets!
Example 5
Fully factorise
Be careful, this one is not the difference of two squares!
We first need to find the greatest common factor (
Write down two brackets with the x at the front
Square root the first term and write it on the left hand side of both brackets
Square root the last term and write it on the right hand side of both brackets
Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.)
We can check the final answer by multiplying out the brackets!
Common Misconceptions
- There must be a + in one bracket, and a – in the other, the order doesn’t matter.
( + )( – ) or( – )( + ) - Remember to square root the entire term including the coefficients of the variables.
E.g.9x2 – 49 √9x2 = 3x √49 = 7 - The term factorising can sometimes be written as ‘factoring’ or factorization’
Practice Factorising Questions: difference of two squares
1. Fully factorise x2 – 25
= (x + 5)(x – 5)
2. Fully factorise: y2 – 81
= (y + 9)(y – 9)
3. Fully factorise: 49 – y2
= (7 + y)(7 – y)
4. Fully factorise: 4 – x2
= (2 + x)(2 – x)
5. Fully factorise: 16x2 – 100
= (4x + 10)(4x -10)
6. Fully factorise: 49 – 9y2
= (7 + 3y)(7 – 3y)
7. Fully factorise: 81x2 – 16y2
= (9x + 4y)(9x – 4y)
8. Fully factorise: 4x3 – 36x
= 4x(x + 3)(x – 3)
GCSE Factorising Questions: difference of two squares
1. Factorise x2 – 100
= (x + 10)(x – 10)
(2 marks)
2. Factorise y2 – 49
= (y + 7)(y – 7)
(2 marks)
3. Factorise 2x2 – 50
= 2(x2 – 25) = 2(x + 5)(x – 5)
(2 marks)
Factorising worksheeets
Download for free one or more factorising worksheets to help you practise this further and more.
Learning Checklist:
You have now learnt to:
- Manipulate algebraic expressions by taking out common factors to factorise into a single bracket.
- Factorise quadratic expressions of the form x2 + bx + c
- Factorise quadratic expressions of the form of the difference of two squares.
- Factorising quadratic expressions of the form ax2 + bx + c (H)
Head back to the main Factorising page for a recap on all types of factorising, along with, mixed practice questions, mixed worksheets and exam questions.
The next topics are:
- Solving Equations
- Solving Linear Equations
- Solving Quadratic Equations
- Quadratic Formula
- Completing the square
- Forming and solving quadratic equations
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