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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:
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.
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)
In order to factorise an algebraic expression using the difference of two squares:
Example 1
Fully factorise
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!
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)
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)
Download for free one or more factorising worksheets to help you practise this further and more.
You have now learnt to:
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:
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