Let be the roots of the equation x 2 - x + 2 = 0 with. Then is equal to ____ . [2024]

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Solution

Q.
Let $α, β$ be the roots of the equation $x^{2} - x + 2 = 0$ with $Im (α) > Im (β) .$ Then $α^{6} + α^{4} + β^{4} - 5 α^{2}$ is equal to ____ . [2024]

Ans.
(13)

We have, $x^{2} - x + 2 = 0$

As $α, β$ are the roots of equation so, it satisfy the equation.

$α^{2} - α + 2 = 0 \Rightarrow α^{2} = α - 2$

$\Rightarrow α^{4} = {(α - 2)}^{2}$ [Squaring both sides]

$\Rightarrow α^{4} = 4 + α^{2} - 4 α = 4 + α - 2 - 4 α = 2 - 3 α$

and $α^{6} = α^{2} \cdot α^{4} = (α - 2) (2 - 3 α) = 2 α - 3 α^{2} - 4 + 6 α$

$= 8 α - 4 - 3 (α - 2) = 8 α - 4 - 3 α + 6 = 5 α + 2$

Similarly, $β^{4} = 2 - 3 β$

Now, $α^{6} + α^{4} + β^{4} - 5 α^{2}$

$= 5 α + 2 + 2 - 3 α + 2 - 3 β - 5 (α - 2)$

$= 16 - 3 (α + β) = 16 - 3 (1) = 13$

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