4 The equation \(x ^ { 3 } - x ^ { 2 } - 6 = 0\) has one real root, denoted by \(\alpha\).
- Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
- Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \sqrt { } \left( x _ { n } + \frac { 6 } { x _ { n } } \right)$$
converges, then it converges to \(\alpha\).
- Use this iterative formula to determine \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.