Newton's Method for Finding Roots of an Equation

You will find that the problem of finding the roots of an equation occurs frequently in technical work. One technique for finding roots is called Newton's Method. That method is the focus of this page.

Assume you have an equation, y = f(x)
• You want the root(s) of the equation so you need find the solution(s) to the equation f(x) = 0
• A root of the equation will occur when an x-coordinate makes the value of the function equal to zero.
• This is also the point where the graph crosses the x axis.

To start the process you will need a first approximation to the desired root (perhaps a guess). You can then use that approximation (guess) as input to the process to develop an improved guess. Through a process of iteration you can get as close to the true root as you desire. This is an iterative process that leads to an approximate solution. If the equation has multiple roots you will have to repeat the process for each root.

An easy way to get a first guess is to graph the function or calculate a set of points in the interval of interest. The bottom part of this page has two tools. First is has a tool that will help you develop a first estimate of a root. Secondly it has a tool that can use your estimate as an input to finding a better estiamte of the root. This second tool uses Newton's Method.

This program will calculate the roots of a quadratic equation of the form f(x) = ax² + bx + c one step at a time. You must take each result and feed it into the program as your new value of x.