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  4. Let α and β be two roots of the equation x2 + 2x + 2 = 0, then α15 + β15 is equal to :

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Q. Let $\alpha$ and $\beta$ be two roots of the equation $x^2 + 2x + 2 = 0$, then $\alpha^{15 } + \beta^{15}$ is equal to :

JEE MainJEE Main 2019Complex Numbers and Quadratic Equations

Solution:

We have
$(x + 1)^2 + 1 = 0$
$\Rightarrow \; (x + 1)^2 - (i)^2 = 0 $
$\Rightarrow \; (x + 1 + i) (x + 1 - i) = 0$
$\therefore \; x = - (1+i) - ( 1 -i)$
$\alpha^{15} + \beta^{15} = \left(\alpha^{2}\right)^{7} \alpha + \left(\beta^{2}\right)^{7} \beta $
= $-128 (-i + 1 + i + 1)$
= $- 256$