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 determined correctly, allowing for any rounding the final answer.


1. Find an approximate solution to the equation `x^3 - 3x - 44 = 0`. Give your answer to 1 decimal place.

Answer: 3.8

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:

452Too high
318Too low
3.532.375Too low
3.843.472Too low
3.947.619Too high
3.8545.517Rounding check

3.8 is too low (gives 43.472); and 3.85 is too high (45.516). Therefore the answer lies above 3.8 and below 3.85, and rounds to 3.8 (1dp).

2. Find a solution, to 1 decimal place, to the equation `x^3-x=100`. Use a startig value of 5.

Answer: 5.4

Create a table for the iteration:

5120Too high
460Too low
4.799.123Too low
4.8105.792Too high
4.75102.422Rounding 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 therefore rounds down to 4.7.