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Gateway to A Level – GCSE
revision
B. Factorising quadratics
You need to be able to factorise quadratic expressions of the form + + .
If the coefficient of is 1, look for a factorisation of the form ( − )( − ). The numbers and
are such that their product is and they add up to .
Worked example 1
Factorise − 7 + 12
Solution Comments
− + Look at factors of . You need two negative
numbers to give + 12.
12 = ( − 1) × ( − 12)
= ( − 2) × ( − 6)
= ( − 3) × ( − 4)
= ( − 3)( − 4) − 3 and − 4 add up to − .
If the coefficient of is not 1 , you need to adapt this procedure slightly. First look for two
numbers that multiply to and add up to . Then split the middle term and factorise in pairs.
Worked example 2
Factorise 6 + 11 − 10
Solution Comments
×(− ) = − 60 You need two numbers that multiply to
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−60 and add up to 11. This means you
need one positive and one negative
number, with the positive number being
larger.
The two numbers are − 4 and 15 Look at factors of 60:
( − 4 + 15 = 11) − 60 = ( − 1) × 60
= ( − 3) × 20
= ( − 4) × 15
6 + − 10 = 6 − + − 10 Split the middle term: 11 = − 4 + 15
Factorise in pairs: The first two terms
= 2 (3 − 2) + 5(3 − 2) have a common factor 2 and the last two
terms have a common factor 5.
= (3 − 2)(2 + 5) Finally, take out the common factor
(3 − 2).
An alternative method is to simply look for numbers that work. If the coefficients are small prime
numbers this can be quite quick, but otherwise the method shown in Worked example 2 is more
efficient.
Worked example 3
Factorise 5 +9 −2
Solution Comments
5 + 9 − 2 = (5 )( ) The only way to factorise 5 is 5 × 1.
= (5 − 1)( + 2) The missing numbers in brackets are 1 and 2.
One is positive one is negative.
Try possible combinations until you find one that
gives the middle term +9 .
Before you use one of these methods, you should check whether there is a common factor that
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can be taken out of all three terms. For example:
7 − 35 + 42 = 7( − 5 + 6)
= 7( − 2)( − 3)
A special example of factorising a quadratic is the difference of two squares:
− = ( − )( + )
Worked example 4
Factorise 9 − 25
Solution Comments
9 − 25 = (3 − 5)(3 + 5) 9 is the square of 3 and 25 is the square of 5.
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EXERCISE B
Factorise these expressions:
1 a +2 −3
b − 2 − 35
c − 8 − 20
d + 13 + 40
2 a 2 − 4 − 30
b 5 + 5 − 30
c 2 − 4 − 16
d 3 − 18 + 27
3 a − 81
b − 100
c 16 − 49
d 36 − 81
4 a 3 − 14 − 5
b 2 − −3
c 5 − 14 − 3
d 2 + − 10
e 6 −5 +1
f 15 + 13 + 2
g 6 − − 15
h 10 + − 21
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