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The diagram shows the curve \(y = x ^ { 4 } - 8 x\).
- By sketching a second curve on the copy of the diagram, show that the equation
$$x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0$$
has two real roots. State the equation of the second curve.
- The larger root of the equation \(x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0\) is denoted by \(\alpha\).
(a) Show by calculation that \(2.1 < \alpha < 2.2\).
(b) Use an iterative process based on the equation
$$x = \sqrt [ 4 ] { 9 + 8 x - x ^ { 2 } } ,$$
with a suitable starting value, to find \(\alpha\) correct to 3 decimal places. Give the result of each step of the iterative process.