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In order to access this I need to be confident with:
Expanding brackets
Factors and multiples
Terms, Expressions, Equations, Formulas
Adding and subtracting Negative numbers
Multiplying Negative Numbers
This topic is relevant for:
Here we will learn about factorising quadratics, including understanding quadratic expressions and the steps needed to factorise into double brackets.
There are also factorising quadratics worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
First of all, let’s have a quick recap on quadratic expressions.
A quadratic expression in maths is an expression including a squared term ie a term up to
The highest power for a quadratic expression is
The general form of a quadratic expression is:
e.g.
We factorise quadratic expressions of this sort using double brackets, the methods are slightly different depending on whether there is a coefficient for x.
A quadratic equation is a quadratic expression that is equal to something. We can solve quadratic equations by using factorisation, the quadratic formula or by using the completing the square.
To factorise a quadratic expression in the form
In this case, the coefficient (number in front) of the
x2 + 6x + 5 = (x + 5)(x + 1)
In order to factorise a quadratic algebraic expression in the form
Factors of 5:
1, 5
2We need a pair of factors that + to give the middle number (6) and ✕ to give the last number (5).
Factors of 5:
1, 5
1 + 5 = 6 ✔
1 ✕ 5 = 5 ✔
(It’s a good idea to do a quick check that we have the correct numbers)
Remember: to ✕ two values together to give a positive answer, the signs must be the same
3Write two brackets and put the variable at the start of each one (x in this case).
4Write one factor in the first bracket and the other factor in the second bracket. The order isn’t important, the signs of the factors are.
We have now fully factorised the quadratic expression.
We can check the answer by multiplying out the brackets!
Write out the factor pairs of the last number (24) in order
Factors of 24:
1, 24
2, 12
3, 8
4, 6
We need a pair of factors that + to give the middle number (-2) and ✕ to give the last number (-24).
Factors of 24:
1, 24
2, 12
3, 8
4, 6
-6 + 4 = -2 ✔
-6 ✕ 4 = -24 ✔
(It’s a good idea to do a quick check that we have the correct numbers)
Remember: to x two values together to give a negative answer, the signs must be the different.
Write two brackets and put the variable at the start of each one (x in this case).
Write one factor in the first bracket and the other factor in the second bracket. The order isn’t important, the signs of the factors are.
(x – 6)(x + 4)
The order of the brackets doesn’t matter
We have now fully factorised the quadratic expression.
We can check the answer by multiplying out the brackets!
Example 3
Fully Factorise:
Write out the factor pairs of the last number (20) in order.
Factors of 20:
1, 20
2, 10
4, 5
We need a pair of factors that + to give the middle number (1) and ✕ to give the last number (-20).
Factors of 20:
1, 20
2, 10
4, 5
-4 + 5 = 1 ✔
-4 ✕ 5 = -20 ✔
(It’s a good idea to do a quick check that we have the correct numbers)
Remember: to ✕ two values together to give a negative answer, the signs must be the different
Write two brackets and put the variable at the start of each one (x in this case)
Write one factor in the first bracket and the other factor in the second bracket. The order isn’t important, the signs of the factors are.
(x – 4)(x + 5)
(The order of the brackets doesn’t matter)
We can check the answer by multiplying out the brackets!
Example 4
Fully factorise:
Write out the factor pairs of the last number (15) in order.
Factors of 15:
1, 15
3, 5
We need a pair of factors that + to give the middle number (-8) and ✕ to give the last number (15).
Factors of 15:
1, 15
3, 5
-3 + -5 = -8 ✔
-3 ✕ -5 = 15 ✔
It’s a good idea to do a quick check that we have the correct numbers.
Remember: to ✕ two values together to give a positive answer, the signs must be the same
Write two brackets and put the variable at the start of each one (x in this case)
Write one factor in the first bracket and the other factor in the second bracket.
(x – 3
)(x – 5
)
(The order of the brackets doesn’t matter)
We have now fully factorised the quadratic expression.
We can check the answer by multiplying out the brackets!
For two numbers to multiply to give a + their signs must be the same
+ ✕ + = +
e.g 2 ✕ 3 = 6
4 ✕ 5=20
– ✕ – = +
e.g -2 ✕ -3 = 6
-4 ✕ -5= 20
For two numbers to multiply to give a – their signs must be the different
+ ✕ – = –
e.g 2 ✕ -3= -6
4 ✕ -5= -20
– ✕ + = –
e.g -2 ✕ 3= -6
-4 ✕ 5= -20
Related content: Negative numbers
1. Fully factorise: x2 + 5x + 6
=(x+3)(x+2)
2. Fully factorise: x2 + 10x + 21
= (x + 3)(x + 7)
3. Fully factorise: x2 – x – 12
=(x – 4)(x + 3)
4. Fully factorise: x2 + 3x – 18
=(x + 6)(x – 3)
5. Fully factorise: x2 – 6x + 8
=(x – 2)(x – 4)
6. Fully factorise: x2 – 10x + 24
=(x – 4)(x – 6)
1. Factorise x2 + 3x – 10
(x – 2)(x + 5)
(2 marks)
2. Factorise y2 – 10y + 16
(y – 2)(y – 8)
(2 marks)
3. Factorise x2 – 12x + 27
(x – 3)(x – 9)
(2 marks)
Download two free factorising quadratics worksheets to help your students prepare for GCSEs.
To factorise a quadratic expression in the form
In this case the coefficient (number in front) of the
2x2 + 5x + 3 = (2x + 3)(x + 1)
In order to factorise a quadratic algebraic expression in the form
Example 1
Fully factorise:
Factors of 6:
1, 6
2, 3
2We need a pair of factors that + to give the middle number (
Factors of 6:
1, 6
2, 3
⊕ 6
✕ 5
2 + 3 = 5 ✔
2 x 3 = 6 ✔
Remember: to x two values together to give a positive answer, the signs must be the same.
3Go back to the original equation and rewrite it this time splitting the middle term into the two factors we found in step 2 – the order of these factors doesn’t matter, the signs do.
2x2 + 5x + 3
2x2 + 2x + 3x + 3
4Split the equation down the middle into two halves and fully factorise each half – the expressions in the brackets must be the same!
2x2 + 5x + 3
2x2 + 2x + 3x + 3
2x(x + 1) + 3(x + 1)
5Now factorise the whole expression by bringing whatever is in the bracket to the front and writing the two other terms in the other bracket.
(x + 1)(2x+ 3)
The order of the brackets doesn’t matter
We have now fully factorised the quadratic expression.
We can check the answer by multiplying out the brackets!
Example 2
Fully factorise
Multiply the the end numbers together (2 and -2) then write out the factor pairs of this new number in order.
Factors of 4:
1, 4
2, 2
We need a pair of factors that + to give the middle number (3) and ✕ to give this new number (-4)
Factors of 4:
1, 4
2, 2
⊕ 3
✕ -4
-1 + 4 = 3 ✔
-1 ✕ 4 = -4 ✔
Remember: to x two values together to give a negative answer, the signs must be different
Go back to the original equation and rewrite it this time splitting the middle term into the two factors we found in step 2 – the order of these factors doesn’t matter, the signs do.
2x2 + 3x - 2
2x2 - x + 4x - 2
Split the equation down the middle into two halves and fully factorise each half – the expressions in the brackets must be the same!
2x2 + 3x - 2
2x2 - x + 4x - 2
x(2x - 1) + 2(2x - 1)
Now factorise the whole expression by bringing whatever is in the bracket to the front and writing the two other terms in the other bracket.
We have now fully factorised the quadratic expression.
We can check the answer by multiplying out the brackets!
Example 3
Fully factorise:
Multiply the the end numbers together (3 and -8) then write out the factor pairs of this new number in order.
Factors of 24:
1, 24
2, 12
3, 8
4, 6
We need a pair of factors that + to give the middle number (-2) and to ✕ give this new number (-24)
Factors of 24:
1, 24
2, 12
3, 8
4, 6
⊕ -2
✕ -24
-6 + 4 = -2 ✔
-6 ✕ 4 = -24 ✔
Remember: to ✕ two values together to give a negative answer, the signs must be different
Go back to the original equation and rewrite it this time splitting the middle term into the two factors we found in step 2 – the order of these factors doesn’t matter, the signs do.
3x2 - 2x - 8
3x2 - 6x + 4x - 8
Split the equation down the middle into two halves and fully factorise each half – the expressions in the brackets must be the same!
3x2 - 2x - 8
3x2 - 6x + 4x - 8
3x(x - 2) + 4(x - 2)
Now factorise the whole expression by bringing whatever is in the bracket to the front and writing the two other terms in the other bracket.
(x – 2)(3x + 4)
(The order of the brackets doesn’t matter)
We have now fully factorised the quadratic expression.
We can check the answer by multiplying out the brackets!
Example 4
Fully factorise:
Multiply the the end numbers together (6 and 2) then write out the factor pairs of this new number in order.
Factors of 12:
1, 12
2, 6
3, 4
We need a pair of factors that + to give the middle number (-7) and ✕ to give this new number (12)
Factors of 12:
1, 12
2, 6
3, 4
⊕ -2
✕ -24
-3 + -4 = -7 ✔
-3 ✕ -4 = 12 ✔
Remember: to ✕ two values together to give a positive answer, the signs must be the same
Go back to the original equation and rewrite it this time splitting the middle term into the two factors we found in step 2 – the order of these factors doesn’t matter, the signs do.
6x2 - 7x + 2
6x2 - 3x - 4x + 2
Split the equation down the middle into two halves and fully factorise each half – the expressions in the brackets must be the same!
6x2 - 7x + 2
6x2 - 3x - 4x + 2
3x(2x - 1) - 2(2x - 1)
Now factorise the whole expression by bringing whatever is in the bracket to the front and writing the two other terms in the other bracket.
(2x – 1)(3x – 2)
(The order of the brackets doesn’t matter)
We have now fully factorised the quadratic expression.
We can check the answer by multiplying out the brackets!
1. Fully Factorise: 2x2 + 5x + 2
=(2x+1)(x+2)
2. Fully Factorise: 2x2 + x – 6
=(2x – 3)(x + 2)
3. Fully factorise: 2x2 – 14x + 20
=(2x – 4)(x – 5)
4. Fully factorise: 3x2 – 7x – 6
=(3x + 2)(x – 3)
5. Fully factorise: 3x2 – 7x + 2
=(3x – 1)(x – 2)
6. Fully factorise: 4x2 – 18x + 8
=(4x – 2)(x – 4)
1. Factorise 2x2 + 9x + 4
(2x + 1)(x + 4)
(2 marks)
2. Factorise 2y2 – y – 3
(2y – 3)(y + 1)
(2 marks)
3. Factorise 2x2 – x – 10
(2x – 5)(x + 2)
(2 marks)
Download two free factorising quadratics worksheets to help your students prepare for GCSEs.
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