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  4. If α and β are roots of the equation x2 + x + 1 = 0 then α2 + β2 is

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Q. If $\alpha$ and $\beta$ are roots of the equation $x^2 + x + 1 = 0$ then $\alpha^2 + \beta^2$ is

KCETKCET 2019Complex Numbers and Quadratic Equations

Solution:

$x^{2}+x+1=0$
$\Rightarrow \, x=\alpha, \beta=\alpha^{2}$
$where\, \alpha\, is\, the\, cube \,root \,of \,unity\,$
$\alpha^{2}+\beta^{2}=\omega^{2}+\omega^{4}$
$=\omega^{2}+\omega$
$=-1$