Transcript File

Trial and Improvement
Objectives:
C Grade
Form and solve equations such as x2 + x = 12
using trial and improvement
Prior knowledge:
•
•
•
•
Rounding to decimal places
Substitution into algebraic expressions
The shape of quadratic / cubic graphs
Use of the bracket button on a calculator
Trial and Improvement
Estimate the square root of the following numbers:
a.
b.
c.
d.
e.
f.
17
30
47
68
110
83
4.1 4.123105626
5.5 5.477225575
6.9 6.8556546
8.2 8.246211251
10.5 10.48808848
9.1 9.110433579
Now check your answers on a calculator
Trial and Improvement
Find the positive solution to the equation
x2 - x = 60
give your answer to 1 decimal place
85
80
y = x2 - x
75
70
65
If we consider this drawing a graph y = 60
we know the solution can be found by
drawing the line y = 60
and y = x2 – x, finding the value of x at
the point of intersection.
60
55
50
45
40
35
30
because of the scale of the graph we
have to use we cannot find the value
of x to 1 d.p. but we can see it is
between 8 and 9.
25
20
15
10
5
0
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-5
Trial and Improvement
Trial and improvement is where we try
values of x in the equation and try to
get as close to the given value for y
as possible
2 – x = 60
the solution is
TryNow
x = we
8 knowxthat
8.25
x =we
8 istrytoo
low
64between
– 8 = 568.2 and 8.3
Try x = 9
81 – 9 = 72
x = 9 is too low
Try x = 8.5
72.25 – 8.5 = 63.75 x = 8.5 is too high
Try x = 8.3
68.89 – 8.3 = 60.59 x = 8.3 is too high
Try x = 8.2
67.42 – 8.2 = 59.04 x = 8.2 is too low
85
80
75
70
65
60
55
50
45
40
35
30
25
20
Now15even the expanding
10 is not big enough
graph
5 for the level
0
accuracy
-5 -4 -3of
-2 -1
0 1 2 3 4 required
5 6 7 8 9
-5
Trial and Improvement
Trial and improvement is where we try
values of x in the equation and try to
get as close to the given value for y
as possible
Now we know that the solution is
between 8.2 and 8.3 we try 8.25
Try x = 8.25
68.0625 – 8.25 = 59.8125 x = 8.25 is too low
We need the value of x to 1 d.p.
We know the solution is now between 8.25 and 8.3.
Try x = 8.3
Nowbeeven
the expanding
8.25
and
8.3
would
rounded
to 8.3
68.89 – Any
8.3 =value
60.59between
x = 8.3 is too high
graph is not big enough
to
1
d.p
Try x = 8.2
for the level
Therefore
to
1
d.p
x
=
8.3
67.42 – 8.2 = 63.75 x = 8.2 is too low
of accuracy required
Trial and Improvement
Find the value of x to 1.d.p to solve this equation:
x3 + x = 12
We can now do this as a table:
Trial Value of x
3
2
2.1
2.2
2.15
2.14
x3
x
x3-x
27
3
30
8
2
10
9.261 2.1 11.36
10.65 2.2 12.85
9.938 2.15 12.09
9.8
2.14 11.94
Comment
Too High
Too Low
Too Low
Too High
Too High
Too Low
Because 2.15 is too high and any number less than 2.15 would be
rounded to 2.1 to 1 d.p.
Finding the answer when x = 2.14 proves that this is correct.
Trial and Improvement
Now do these:
Find the positive solutions to 1 decimal place
1. x2 + 2x = 63
2. x2 - 2x = 675
3. x3 + 2x = 520
4. x5 + x = 33 768
5.
x2 - 7x = 368
6.
x - x3 = -336
Trial and Improvement
Worksheet
x2 + 2x = 63
Trial Value of x
x2 - 2x = 675
Trial Value of x
x3 + 2x = 520
Trial Value of x
x2
x2
x3
2x
2x
2x
x2 + 2x Comment
x5 + x = 33 768
Trial Value of x
x5
x
x5 + x Comment
x2 - 2x Comment
x2 - 7x = 368
Trial Value of x
x2
7x
x2 - 7x Comment
x3 + 2x Comment
x - x3 = -336
Trial Value of x
x
x3
x - x3 Comment
Trial and Improvement
x2 + 2x = 63
Trial Value of x
5
6
7
x=7
63
x2
25
36
49
x2 - 2x = 675
Trial Value of x
20
30
28
26
27
x = 27
675
x2
400
900
784
676
729
2x
10
12
14
2x
40
60
56
52
54
x520
= 8.0
x3 + 2x = 520
x3
2x
Trial Value of x
8
512
16
7
343
14
7.5
421.9
15
7.9
493
15.8
7.95
502.5 15.9
7.96
504.4 15.92
x + 2x Comment
35
Too Low
48
Too Low
63
2
2
x - 2x
360
840
728
624
675
x3 + 2x
528
357
436.9
508.8
518.4
520.3
Comment
Too Low
Too High
Too High
Too Low
Comment
Too High
Too Low
Too Low
Too Low
Too Low
Too High
x5 + x = 33 768
Trial Value of x
8
9
8.1
8.05
8.02
8.04
33768
x = 8.0
x5
x
32768 16
59049 18
34868 16.2
33805 16.1
33180 16.04
33595 16.08
x5 + x Comment
32784 Too Low
59067 Too High
34884 Too High
33821 Too High
33196 Too Low
33612 Too Low
x2 - 7x = 368
Trial Value of x
25
22
23
368
x = 23
x2
7x
625
175
484
154
529
161
x2 - 7x Comment
450 Too High
330 Too Low
368
x = 7.3
x - x3 = -336
-336
x
x3
Trial Value of x
7
49
343
8
64
512
7.5
56.25 421.9
7.4
54.76 405.2
7.3
53.29 389
x - x3 Comment
-294 Too High
-448 Too Low
-366 Too Low
-350 Too Low
-336