# Solving equations by trial and improvement

**Solving equations by trial and improvement**

Another method of solving equations can be through trial and improvement. In these scenarios, an approximate answer is provided and the solution is found through trial and improvement.

For example, the equation *x*³ – 3*x* =1 has a solution approximately equal to 1.9. The solution is required to 3 decimal places.

The figures of 1.9 and 1.8 is substituted into the equation as follows:

When *x* = 1.9, *x*³ – 3*x = *1.159 (too high)

When *x* = 1.8, *x*³ – 3*x = *0.432 (too low)

There the solution is between 1.8 and 1.9, so further solutions are trialled:

When *x* = 1.85, *x*³ – 3*x = *0.781625 (too low)

When *x* = 1.87, *x*³ – 3*x = *0.929203 (too low)

When *x* = 1.88, *x*³ – 3*x = *1.004672 (too high)

The solution is between 1.87 and 1.88 therefore the following is trialled:

When *x* = 1.879, *x*³ – 3*x = *1.0008718 (the solution to 3 decimal places)