## Trial and Improvement

Sometimes it happens that we can't factorise an expression or even use the quadratic formula to solve it, for example If we try and solve which is equivalent to we 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 of that 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 of that satisfies the equation is between 1 and 2.We can draw up the table:     Too Big TBToo 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 for If you guess a value for and the answer is too small, increase the size of your guess for If you guess gives an answer that is too big, guess a smaller value for Eventually 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 of to 1 decimal place.

We have to choose between 1.8 and 1.9, and we do this by trying This gave an answer that was too small, so we take the bigger value for and to 1 decimal place the solution to the equation is If our answer for had turned out to be to big, we would have chosen the smaller value for  