6 The equation \(6 \arcsin ( 2 x - 1 ) - x ^ { 2 } = 0\) has exactly one real root.
- Show by calculation that the root lies between 0.5 and 0.6 .
In order to find the root, the iterative formula
$$x _ { n + 1 } = p + q \sin \left( r x _ { n } ^ { 2 } \right)$$
with initial value \(x _ { 0 } = 0.5\), is to be used.
- Determine the values of the constants \(p , q\) and \(r\).
- Hence find the root correct to \(\mathbf { 4 }\) significant figures. Show the result of each step of the iteration process.