This by working out values of x

This method works by drawing a graph, seeing where it crosses the axis, then re-plotting the graph, with a smaller scale. This is repeated until required accuracy is achieved. I require an accuracy of four decimal places. The equation I shall investigate will be: y = x4+2×3-5×2+1 I shall find the root by working out values of x and looking for a change of sign. I can see by looking at the graph that the change of sign occurs between 3 and 4. I shall plot a graph, to find a more accurate estimate.

This method requires 7 steps, not including the work put in to rearranging the equation. It is clear to see that decimal search is the simplest way to solve equations. It is however the slowest, as it a table of values has to be made each time, to improve accuracy and the equation has to be written into Excel, which can take time. Doing it by hand would also take a long time, as you would have to put numbers through the equation up to ten times before the change of sign is found. This will be very monotonous and time consuming.

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The Newton-Raphson method is the fastest, especially if you have a computer with Autograph to draw the graphs and to the maths behind it for you. A few more steps are necessary if you use the formula each time to improve your answer. The rearrangement method required the largest number of steps, and on top of that, the equation had to be rearranged twice. Overall, I believe the Newton-Raphson method is the best method to use when solving equations, because it is the shortest by hand and without a doubt the shortest and easiest using a computer.