The word 'iterate' means to do something repeatedly, with a slight improvement each time. In mathematics, an iterative method is one which gradually gets us closer to a solution.

In this tutorial, we'll be looking at a clever type of iteration which helps us to solve equations that are hard to solve using algebra alone.

To do this, we start with some value $x_0$ which we believe to be close to the correct answer. We then iterate to find $x_1$ and then iterate again to find $x_2.$ Generally, $x_{n+1}$ should be closer to the real solution than $x_n.$

Let's see how this is done!

1. Use an iterative method, with a starting value of $x_0=1.4,$ to find $2$ further approximations for one of the roots of

   $$3x^2+x-7=0$$

2. Find the value of the root from (a) to $5$ decimal places.

3. Draw a diagram to demonstrate the iterative process used in (a) and (b).