Answer
$x = 2.236067997$
Work Step by Step
$f(x) = x^2 - 5$ and $x_1 = 2.2$
1.) Use the formula $x_{n + 1} = x_n - \dfrac{f(x)}{f'(x)}$ to find $x_2$
$x_2 = 2.2 - \dfrac{f(2.2)}{f'(2.2)} = 2.2 - \dfrac{-0.16}{4.4}$ = 2.2363636
2.) Use the same formula to calculate $x_3$
$x_3 = x_2 - \dfrac{f(2.2363636)}{f'(2.2363636)} = 2.2363636 - \dfrac{0.001322314}{4.72727273}$ = 2.236067997
We have now completed two iterations of Newton's Method, and have found the approximation to be
x = 2.236067997
