7 The equation
$$24 x ^ { 3 } + 36 x ^ { 2 } + 18 x - 5 = 0$$
has one real root, \(\alpha\).
- Show that \(\alpha\) lies in the interval \(0.1 < x < 0.2\).
- Starting from the interval \(0.1 < x < 0.2\), use interval bisection twice to obtain an interval of width 0.025 within which \(\alpha\) must lie.
- Taking \(x _ { 1 } = 0.2\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to four decimal places.
(4 marks)