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13.5 Trial and improvement

13.5 Trial and improvement

Look at these three equations.
t 2x + 3 = 28
!e solution of this equation is x = (28 − 3) ÷ 2 = 12.5
t x + 3x = 28
2
You cannot solve this by rearranging the terms. One way to solve it is to try di%erent values of x.
A solution is x = 4 because 4 + (3 × 4) = 16 + 12 = 28
2
t x + Y = 36
2
Again, you cannot solve this by rearranging the terms. Try di%erent values of x.
If x = 4, x + 3x = 28 !is is too small.
2
If x = 5, x + 3x = 5 + (3 × 5) = 40 !is is too large.
2
2
Try a value between 4 and 5. Try x = 4.5.
If x = 4.5, x + 3x = 4.5 + (3 × 4.5) = 33.75 !is is too small.
2
2
Try a value between 4.5 and 5. Try 4.6.
If x = 4.6, x + 3x = 4.6 + (3 × 4.6) = 34.96 !is is too small.
2
2
Try 4.7 x value
If x = 4.7, x + 3x = 4.7 + (3 × 4.7) = 36.19 !is is too large. 4 28 too small
2
2
!is method is called trial and improvement. 5 40 too large
4.5 33.75 too small
You try to get closer and closer to the exact answer. 4.6 34.96 too small
!e table on the right gives answers closer and closer to 36. 4.7 36.19 too large

4.65 was chosen because it is halfway between 4.6 and 4.7. 4.65 35.5725 too small
!e exact answer is between 4.68 and 4.69. 4.68 35.9424 too small
!e answer, to one decimal place, is 4.7. 4.69 36.0661 too large

Worked example 13.5

Use trial and improvement to find a positive solution to the equation x(x − 2) = 60.
Give the answer correct to one decimal place.

The table shows the values tried.
x x(x − 2)
6 6 × 4 = 24 too small It is a good idea to put the results in a table.
8 8 × 6 = 48 too small The value of x is between 8 and 9. It is closer to 9.
9 63 too large The value of x is between 8.8 and 8.9.
8.8 59.84 too small
8.8 is closer than 8.9.
8.9 61.41 too large
The solution, to one decimal place, is x = 8.8.
8.85 60.6225 too large

130 13 Equations and inequalities