10. The equation
$$1 + 5 x - x ^ { 4 } = 0$$
has a positive root \(\alpha\).
- Show that \(\alpha\) lies between 1 and 2 .
- Use the iterative sequence based on the arrangement
$$x = \sqrt [ 4 ] { 1 + 5 x }$$
with starting value 1.5 to find \(\alpha\) correct to two decimal places.
- Use the Newton-Raphson method to find \(\alpha\) correct to six decimal places.