Let be the roots of the equation x 2 - 2 x + 2 = 0 . Then 14 +14 is equal to [2023]

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Solution

Q.
Let $α, β$ be the roots of the equation $x^{2} - \sqrt{2} x + 2 = 0 .$ Then $α^{14} + β^{14}$ is equal to [2023]

1 $- 64 \sqrt{2}$

2 $- 128$

3 $- 128 \sqrt{2}$

4 $- 64$

Ans.
(2)

$Given, x^{2} - \sqrt{2} x + 2 = 0$

$x = \frac{\sqrt{2} \pm \sqrt{{(- \sqrt{2})}^{2} - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

$x = \frac{\sqrt{2} \pm \sqrt{- 6}}{2};$ $x = \frac{1 \pm \sqrt{3} i}{\sqrt{2}}$ $\Rightarrow α = - \sqrt{2} ω, β = - \sqrt{2} ω^{2}$

$\Rightarrow α^{14} + β^{14} = {(- \sqrt{2} ω)}^{14} + {(- \sqrt{2} ω^{2})}^{14}$

$= 2^{7} \cdot ω^{14} + 2^{7} ω^{28} = 2^{7} (ω + ω^{2})$

$= - 2^{7}$ $(∵ ω^{2} + ω + 1 = 0)$

$= - 128$

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