The Idea of Newton's Method

Newton's method is a technique for solving equations of the form $f(x)=0$ by successive approximation. The idea is to pick an initial guess $x_0$ such that $f(x_0)$ is reasonably close to 0. We then find the equation of the line tangent to $y=f(x)$ at $x=x_0$ and follow it back to the $x$ axis at a new (and improved!) guess $x_1$. The formula for this is $$ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}.$$ We then find the equation of the line tangent to $y=f(x)$ at $x=x_1$ and follow it back to the $x$ axis to get a new (and improved!) guess $x_2$ from the formula $$ x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}.$$

We keep on refining our guesses until we are close enough for whatever application we have in mind. In general, we have the recursive formula

In typical situations, Newton's method homes in on the answer extremely quickly, roughly doubling the number of decimal points in each round. So if your original guess is good to one decimal place, 5 rounds later you will have an answer good to 30+ digits.