The Newton-Raphson method is another way for solving an equation of the form

Suppose $\alpha$ is a solution to this equation, and $x_0$ is a number we believe to be close to $\alpha .$ The image below shows how the Newton-Raphson method works:

The idea is:

• We guess an $x_0$ and find the point $(x_0,y_0)$ on the curve

• We find the equation of the tangent to the curve at $(x_0,y_0)$

• Then $x_1$ is given by the point where this tangent crosses the $x$ axis

Now, that process is a little long-winded. Using algebra, we can show that

This can be done again and again to get closer and closer to $\alpha .$ Generally,

Use the Newton-Raphson method, with a starting value of $x_0=1.1$ to find $2$ further approximations for one of the solutions to