Trial and Improvement
1
Trial and Improvement - Foundation Tier
In today’s lesson you will:
Understand that some equations are
difficult (or even impossible!) to solve
using the techniques you already know
[ALL of you];
 Learn how to get a rough (approximate)
answer to an equation using the method
known as trial and improvement
[ALL of you];
 Use this method on more difficult
problems
[MOST of you].

2
Trial and Improvement - Foundation Tier
The basic idea of T & I ……
We will start by using Trial and Improvement
on a problem that we can actually solve
using algebra:
Solve the equation:
20x + 16 = 60
3
Trial and Improvement - Foundation Tier
The basic idea of T & I ……2
We begin by guessing what we think the answer to
the equation 20x + 16 = 60 might be. So choose
a reasonable guess (or trial) for x.
Let’s try x = 4
So work out the part of the equation with x in it (the
left hand side), changing x to equal 4:
We get:
20 x 4 + 16
Now this equals 80 + 16 = 96. In the equation we
need it to equal 60. This means our original guess
for x was too big…….
So try a smaller guess…….maybe x = 3 ?
4
Trial and Improvement - Foundation Tier
The basic idea of T & I ……3
Now it is a lot easier to put all our working into a
table like this:
20x+16=60
Trial x
20x + 16
Too high/too low
4
80+16 = 96
Too High
3
5
Trial and Improvement - Foundation Tier
The basic idea of T & I ……5
Now we will use the method on a problem
that we would find much more difficult to
solve using algebra:
Solve the equation:
2
x
+ x = 26
Correct to 1 d.p.
6
Trial and Improvement - Foundation Tier
The basic idea of T & I ……6
It’s much easier to put all our working into this
table:
x2 + x = 26
Trial x
x2 + x
Too high/too low
4
7
Trial and Improvement - Foundation Tier
The basic idea of T & I ……7
x=4
is a good guess to begin with:
x2 + x = 26
x2 + x
Trial x
Too high / too low
4
16 + 4 = 20
Too low
5
25 + 5 = 30
Too high
4.5
20.25 + 4.5 = 24.75
Too low
4.6
21.16 + 4.6 = 25.76
Too low
4.7
22.09 + 4.7 = 26.79
Too high
4.65
21.6225 + 4.65 = 26.2725
Too high
8
Trial and Improvement - Foundation Tier
The basic idea of T & I ……7
x2 + x = 26 (to 1 d.p.)
So our guess for x is now between 4.6 and 4.65 –
this means it could be:
4.61 or 4.62 or 4.63 or 4.64 or anything in
between…..
But all of these will be 4.6 if rounded to 1 d.p.
So x = 4.6 is our answer.
Trial x
4.6
4.7
4.65
x2 + x
Too high / too low
21.16 + 4.6 = 25.76
Too low
22.09 + 4.7 = 26.79
Too high
21.6225 + 4.65 = 26.2725
Too high
9
Trial and Improvement - Foundation Tier
The basic idea of T & I ……8
Now you will use the method on a problem
similar to the previous one:
Solve the equation:
x2 – x = 66
Correct to 1 d.p.
10
Trial and Improvement - Foundation Tier
The basic idea of T & I ……9
It will be much easier to put all your working into
this table:
x2 – x = 66
Trial x
x2 – x
Too high/too low
10
11
Trial and Improvement - Foundation Tier
The basic idea of T & I ……10
The first three lines of your working should be:
x2 – x = 66
Trial x
x2 – x
Too high/too low
10
100-10 = 90
Too high
9
81 – 9 = 72
Too high
8
64 – 8 = 56
Too low
8.5
12
Trial and Improvement - Foundation Tier
The basic idea of T & I ……11
x2 - x = 66 (to 1 d.p.)
Your last guess for x should be between 8.6
and 8.65 – this means it could be:
8.61 or 8.62 or 8.63 or 8.64 or anything in
between…..
But all of these will be 8.6 if rounded to 1
d.p.
So x = 8.6 is our answer.
13
Trial and Improvement - Foundation Tier
The basic idea of T & I ……12
Now you will use the method on another
problem:
Solve the equation:
x2 + 2x = 58
Correct to 1 d.p.
14
Trial and Improvement - Foundation Tier
In today’s lesson you should have:
Understood that some equations are
difficult (or even impossible!) to solve
using the techniques you already know
[ALL of you];
 Learnt how to get a rough (approximate)
answer to an equation using the method
known as trial and improvement
[ALL of you];
 Used this method on more difficult
problems
[MOST of you].

15
Trial and Improvement - Foundation Tier
Typical GCSE Exam question:
Use the method of trial and improvement to
find a solution, to 1 decimal place, of the
equation x³ = 100.
(4 marks)
Trial, x
x3
Too high/too low
5
5x5x5 = 125
Too high
4
4x4x4 = 64
Too low
16
Trial and Improvement - Foundation Tier