Therefore,
6
x
2
+
x
– 12 =
6
x
2
- 8
x
+ 9
x
– 12
= 2
x
(3
x
– 4) + 3(3
x
– 4)
(the two brackets must be identical)
= (3
x
– 4)(2
x
+ 3)
Difference of two squares:
Factorising quadratics of the form
2
2
x
a
Remember that
2
2
x
a
= (
x + a
)(
x – a
).
Therefore:
2
2
2
9
3
(
3)(
3)
x
x
x
x
and
2
2
2
16
25
(2 )
5
(2
5)(2
5)
x
x
x
x
Also notice that:
2
2
2
8
2(
4)
2(
4)(
4)
x
x
x
x
and
3
2
2
2
3
48
3 (
16
)
3 (
4 )(
4 )
x
xy
x x
y
x x
y
x
y
Factorising by pairing
We can factorise expressions like
2
2
2
x
xy
x
y
using the method of factorising by pairing:
2
2
2
x
xy
x
y
=
x
(2
x
+
y
) – 1(2
x
+
y
)
(factorise front and back pairs, ensuring both
brackets are identical)
17

= (2
x
+
y
)(
x
– 1)
Exercise B
Factorise
1)
2
6
x
x
2)
2
6
16
x
x
3)
2
2
5
2
x
x
4)
2
2
3
x
x
(factorise by taking out one common factor)
5)
2
3
5
2
x
x
6)
2
2
17
21
y
y
7)
2
7
10
3
y
y
8)
2
10
5
30
x
x
9)
2
4
25
x
10)
x
2
−
3
x
−
xy
+
3
y
11)
2
4
12
8
x
x
12)
2
2
16
81
m
n
13)
3
2
4
9
y
a y
18

14)
2
8(
1)
2(
1)
10
x
x
(expand first)
19

Chapter 5:
CHANGING THE SUBJECT OF A FORMULA
We can use algebra to change the subject of a formula.
Rearranging a formula is similar to
solving an equation – we must do the same to both sides in order to keep the equation balanced.
Example 1
:
Make
x
the subject of the formula
y
= 4
x
+ 3.
Solution
:
y
= 4
x
+ 3
Subtract 3 from both sides:
y
– 3 = 4
x
Divide both sides by 4;
3
4
y
x
So
3
4
y
x
is the same equation but with
x
the subject.
Example 2
:
Make
x
the subject of
y
= 2 – 5
x
Solution
:
Notice that in this formula the
x
term is negative.
y
= 2 – 5
x
Add 5
x
to both sides
y
+ 5
x
= 2
(the
x
term is now positive)
Subtract
y
from both sides
5
x
= 2 –
y
Divide both sides by 5
2
5
y
x
Example 3
: The formula
5(
32)
9
F
C
is used to convert between ° Fahrenheit and ° Celsius.
We can rearrange to make
F
the subject.
5(
32)
9
F
C
Multiply by 9
9
5(
32)
C
F
(this removes the fraction)
Expand the brackets
9
5
160
C
F
Add 160 to both sides
9
160
5
C
F
Divide both sides by 5
9
160
5
C
F
Therefore the required rearrangement is
9
160
5
C
F
Exercise A
Make
x
the subject of each of these formulae:
1)
y
= 7
x
– 1
2)
5
4
x
y
3)
4
2
3
x
y
4)
4(3
5)
9
x
y
20

21

Rearranging equations involving squares and square roots
Example 4
:
Make
x
the subject of
2
2
2
x
y
w
Solution
:
2
2
2
x
y
w
Subtract
2
y
from both sides:
2
2
2
x
w
y
(this isolates the term involving
x
)
Square root both sides:
2
2
x
w
y
Remember that you can have a positive or a negative square root
.
We cannot simplify the answer any more.
Example 5
:
Make
a
the subject of the formula
1
5
4
a
t
h
Solution
:
Multiply by 4
5
4
a
t
h
Square both sides
2
5
16
a
t
h
remember to square the 4
Multiply by
h
:
2
16
5
t h
a
Divide by 5:
2
16
5
t h
a
Exercise B:
Make
t
the subject of each of the following
1)
32
wt
P
r
2)
2
32
wt
P
r
3)
2
1
3
V
t h
4)
2
t
P
g
5)
(
)
w v
t
Pa
g
6)
2
r
a
bt
22

More difficult examples
Sometimes the variable that we wish to make the subject occurs in more than one place in the
formula.
In these questions, we collect the terms involving this variable on one side of the
equation, and we put the other terms on the opposite side.

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- Fall '13
- Quadratic equation, Elementary algebra