# What is Newton's Method?

Newton's Method is a mathematical tool often used in numerical analysis, which serves to approximate the zeroes or roots of a function (that is, all $x : f \left(x\right) = 0$).
The method is constructed as follows: given a function $f \left(x\right)$ defined over the domain of real numbers $x$, and the derivative of said function ($f ' \left(x\right)$), one begins with an estimate or "guess" as to where the function's root might lie. For example, suppose one is presented with the function $f \left(x\right) = {x}^{2} + x - 2.5$. This is similar to another function $g \left(x\right) = {x}^{2} + x - 2$, whose roots are $x = 1$ and $x = - 2$. Thus, thanks to this similarity, one might use $x = 1$ or $x = - 2$ as guesses to start Newton's Method with f(x).
Whatever method used, we declare this initial guess to be ${x}_{0}$. We arrive at a better approximation, ${x}_{1}$, by employing the Method: ${x}_{1} = {x}_{0} - f \frac{{x}_{0}}{f ' \left({x}_{0}\right)}$. Essentially, by utilizing the derivative, one is able to increment closer to the actual value. In the above example, $f \left(x\right) = {x}^{2} + x - 2.5$, if we assume ${x}_{0} = 1$, then ${x}_{1} = 1 - f \frac{1}{f ' \left(1\right)} = 1 - \frac{- .5}{3} = \frac{7}{6} \mathmr{and} \approx 1.16667$.
Often, one may be able to find the root another way (by using a graphing calculator, for example), and an exam item or textbook problem may demand a certain degree of accuracy (such as within 1% of the actual value). In such a case, if ${x}_{1}$ is not an accurate enough approximation, one performs the iteration again, as often as needed for the desired degree of accuracy. The formula to find the general ${x}_{n}$, then, is ${x}_{n} = {x}_{n - 1} - f \frac{{x}_{n - 1}}{f ' \left({x}_{n - 1}\right)}$