Question 1
LEVEL 4
Factorise x^2 + 4x + 3.
Select the correct answer from the options below.
A: (x + 1)(x + 3)
B: (x - 1)(x - 3)
C: (x + 4)(x - 1)
D: (x + 4)(x + 1)
CORRECT ANSWER: A: (x + 1)(x + 3)
WORKED SOLUTION:
We are looking for two numbers which multiply together to make 3 and add to make 4.
We see 1, 3 and -1, -3 are the only pairs of factors of 3,
and 1 + 3 = 4 so the correct pair is 1 and 3.
(x + 1)(x + 3)
Question 2
LEVEL 4
Fully factorise the following quadratic.
x^2 - x - 6
Select the correct answer from the options below:
A: (x + 3)(x - 2)
B: (x - 3)(x + 2)
C: (x - 1)(x + 6)
D: (x + 1)(x - 6)
CORRECT ANSWER: B: (x - 3)(x + 2)
WORKED SOLUTION:
We are looking for two numbers which multiply to make -6 and add to make -1.
The possible factors are:
-6 and 1
-1 and 6
-2 and 3
3 and -2
We can see that -3 + 2 = -1,
so this must be the correct pair. Thus, our factorisation is:
(x - 3)(x + 2)Question 3
LEVEL 4
Fully factorise the following quadratic.
m^2 + m - 42
Select the correct answer from the options below:
A: (m + 7)(m - 6)
B: (m + 3)(m - 14)
C: (m - 3)(m + 14)
D: (m + 6)(m - 7)
CORRECT ANSWER: A: (m + 7)(m - 6)
WORKED SOLUTION:
We are looking for two numbers which multiply to make 42 and add to make 1.
There are lots of possible factorisations, but as they must add to make 1,
we should only consider the factor pairs that are closer together, i.e. 6, -7 and -6, 7.
(m + 7)(m - 6)Question 4
LEVEL 4
Fully factorise the following quadratic.
x^2 - 16
Select the correct answer from the options below:
A: (x - 2)(x + 8)
B: (x + 2)(x - 8)
C: (x - 4)(x - 4)
D: (x - 4)(x + 4)
CORRECT ANSWER: D: (x - 4)(x + 4)
WORKED SOLUTION:
This is a case of the difference of two squares.
Being familiar with these examples can save you time,
as you can immediately determine that because \sqrt{16} = 4
so the factorisation is (x - 4)(x + 4).
This only works when you can square root the number on its own,
so watch our for the squared numbers as this will help you spot
this type of example.
Question 5
LEVEL 4
Fully factorise the following quadratic.
y^2 - 17y + 30
Select the correct answer from the options below:
A: (y - 15)(y + 2)
B: (y - 3)(y - 10)
C: (y - 15)(y - 2)
D: (y + 3)(y - 10)
CORRECT ANSWER: C: (y - 15)(y - 2)
WORKED SOLUTION:
We are looking for two numbers which add to make -17 and multiply to make 30.
30 = -1 \times -30 = -3 \times -10 = -2 \times -15.
Noticing that -2 + (-15) = -17,
we see this must be the correct pairing. So the factorisation is
(y - 15)(y - 2).