If $α, β$ are the roots of the equation $x^{2} - 2 x + 4 = 0$ , then the value of $α^{6} + β^{6}$ is

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Solution

Let $f (x) = x^{2} - 2 x + 4 = 0$ has roots $α, β$ , then the equation whose roots are $α^{6}, β^{6}$ is
$f (x^{1 / 6}) = 0 \Rightarrow x^{1 / 3} - 2 x^{1 / 6} + 4 = 0 \Rightarrow x^{1 / 6} (x^{1 / 6} - 2) = - 4$
Cubing on both sides,
$x^{1 / 2} (x^{1 / 2} - 8 - 6 x^{1 / 6} (x^{1 / 6} - 2)) = - 64 \Rightarrow x^{1 / 2} (x^{1 / 2} - 8 + 24) = - 64 \Rightarrow x + 16 \sqrt{x} = - 64 \Rightarrow x + 64 = - 16 \sqrt{x}$
Squaring on both the sides,
$x^{2} + 128 x + 64^{2} = 256 x \Rightarrow x^{2} - 128 x + 64^{2} = 0$
Therefore, sum of roots is,
$α^{6} + β^{6} = 128$

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