The graph is as follows:

Try x₁ = 1.5

Then

Now f(1.5) = 2(1.5)² − 1.5 − 2 = 1

f '(x) = 4x − 1 and f '(1.5) = 6 − 1 = 5

So

So 1.3 is a better approximation.

Continuing the process,

We can continue for as many steps as necessary to give the required accuracy.

Check: Using some Computer Algebra Systems, (eg Mathcad) we also need to give an initial guess (say, x:=2) and the result is:

root(2x² − x − 2, x) = 1.2807764064044