Verify by calculation that the cubic equation
$$x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3 = 0$$
has a root that lies between \(x = 0.7\) and \(x = 0.8\).
Show that this root also satisfies an equation of the form
$$x = \frac { a x ^ { 2 } + 3 } { x ^ { 2 } + b }$$
where the values of \(a\) and \(b\) are to be found.
With these values of \(a\) and \(b\), use the iterative formula
$$x _ { n + 1 } = \frac { a x _ { n } ^ { 2 } + 3 } { x _ { n } ^ { 2 } + b }$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.