## Trial and Improvement

Sometimes it happens that we can't factorise an expression or even use the quadratic formula to solve it, for exampleIf we try and solvewhich is equivalent towe cannot do it by these methods. However if we know that the answer is in a certain range we can keep making educated guesses until we get the answer that we know is correct say to one decimal place.

We want to solve

We can guess a value ofthat satisfies this equation, say

This is too small so x is probably bigger. We can “improve” our guess to

This is too small so now we can conclude the true value ofthat satisfies the equation is between 1 and 2.We can draw up the table:

Too Big TB or Too Small - TS | |||||

1 | 1 | -2 | -2 | -3 | TS |

2 | 8 | -4 | -2 | 2 | TB |

1.5 | 3.375 | -3 | -2 | -1.125 | TS |

1.7 | 4.913 | -3.4 | -2 | -0.487 | TS |

1.9 | 6.859 | -3.8 | -2 | 1.059 | TB |

1.8 | 5.832 | -3.6 | -2 | 0.232 | TS |

1.85 | 6,331 | -3.7 | -2 | 0.632 | TS |

The procedure is to use the answer from your last guess to try and get a better value forIf you guess a value forand the answer is too small, increase the size of your guess forIf you guess gives an answer that is too big, guess a smaller value forEventually you will find as we did here with 1.8 and 1.9 that one is too small and one is too big but we can't get any closer by guessing values ofto 1 decimal place.

We have to choose between 1.8 and 1.9, and we do this by tryingThis gave an answer that was too small, so we take the bigger value forand to 1 decimal place the solution to the equationisIf our answer forhad turned out to be to big, we would have chosen the smaller value for