This worksheet is about finding approximate solutions to more complex equations using trial and improvement.

We test a value which is near the solution and then home in to a more accurate solution until we reach the desired level of accuracy.

**Example**

One solution to the following equation lies between x = 3 and x = 4.

x^{3} + 2x^{2} + 3x = 100

Using trial and improvement, find this solution to 1 d.p.

x | x^{3} + 2x^{2} + 3x |
Comment |
---|---|---|

3 | 3^{3} + 2×3^{2} + 3×3 = 54 |
too small |

4 | 4^{3} + 2×4^{2} + 3×4= 108 |
too big |

3.5 | 3.5^{3} + 2×3.5^{2} + 3×3.5= 77.875 |
too small |

3.8 | 3.8^{3} + 2×3.8^{2} + 3×3.8= 95.152 |
too small |

3.9 | 3.9^{3} + 2×3.9^{2} + 3×3.9= 101.439 |
just too big |

3.85 | 3.85^{3} + 2×3.85^{2} + 3×3.85= 98.261625 |
just too small |

This shows that x = 3.85 is just too small but x = 3.9 is just too big, so the solutions lies between 3.85 and 3.9.

So to 1 decimal place, we can be sure that it will be x = 3.9 because anything above 3.85 will automatically round up to 3.9 anyway.

So** x = 3.9** (to 1 d.p.)