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Approximate Solutions using Iteration

Approximate Solutions using Iteration

Approximating Solutions, also called Trial and Error, or Trial and Improvement, is used for calculating values when an equation cannot be solved using another method.

The process involves estimating a start value, deriving the answer from the equation, and then improving the next estimate. This process is repeated until the required accuracy is achieved.

The working is normally set out in a table with three columns: the value being tested; the calculation with that value; and a comment about the result.

Check that the accuracy of the solution is correct.

Example 1

Find an approximate solution to the equation `x^3 - 3x - 44 = 0`.

Give your answer to 1 decimal place.

Re-arrange the equation to have the value on one side, and the unknowns on the other: `x^3 - 3x = 44`.

Create a table for the iteration:

Trial Calculation Notes
4 52 Too high
3 18 Too low
3.5 32.375 Too low
3.8 43.472 Too low
3.9 47.619 Too high
3.85 45.517 Rounding check

3.8 is too low (gives 43.472); and 3.85 is too high (45.516).

The answer lies above 3.8 and below 3.85, and rounds to 3.8 (1dp).

Answer: 3.8

Example 2

Find a solution, to 1 decimal place, to the equation `x^3-x=100`.

Use a starting value of 5.

Create a table for the iteration:

Trial Calculation Notes
5 120 Too high
4 60 Too low
4.7 99.123 Too low
4.8 105.792 Too high
4.75 102.422 Rounding check

4.7 is too low (99.123); and 4.75 (105.795) is too high.

The solution is above 4.7 and below 4.75, and rounds down to 4.7.

Answer: 5.4