# Find methods of solving equations, which can't be solved algebraically.

Frayer Walker

SF2

Pure 2 Coursework

The purpose of this investigation is to find methods of solving equations, which can’t be solved algebraically. There are three methods, which can be used to do this: decimal search, Newton-Raphson and rearranging f(x)=0 in the form x=g(x). Then evaluate the usefulness of each method by analysing disadvantages and advantages of each. Finally the best method will be identified.

## Decimal Search

This method uses the fact that as the curve passes through the x-axis there is a change of sign, to identify roots of a chosen function. The fact that there is a change of sign means that the root can be identified to a chosen degree of accuracy in this case four decimal places. This is done by taking increments in x of size 0.1 within an interval containing a root and working out the value of the function for each one. When there is a change of sign this can be repeated between the two value at which it occurs using increments of 0.01 etc. this method is to be used to find the roots for the function x3-3x2+1. The roots, which are to be, identified lie within the intervals [0,1], [-1,0] and [2,3] as shown on the below graph.

The root can now be found using the above method as shown below in the tables.

[0,1]

Therefore it is clear that the error bounds for this root are [0.6527,0.6528] as such the solution is 0.65275 and the maximum error is +0.00005 or –0.00005.

[-1,0]

Therefore it is clear that the error bounds for this root are [-0.5321, -0.5320] as such the solution is –0.55325 and the maximum error is +0.00005 or –0.00005.

[2,3]

Therefore it is clear that the error bounds for this root are [2.8793, 2.8794] as such the solution is 2.87935 and the maximum error is +0.00005 or –0.00005.

However in a number of cases the change of sign method will not work as shown below.

This method also does not work when there are several roots, which are extremely close together. In this case it will ...