Question

Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}+x+1=0$. If the equation whose roots are $\alpha^{58}$ and $\beta^{19}$ is $a x^{2}+b x+c=0$, then find the value of $a+b+c$.

Answer 1

$x^{2}+x+1=0$
$\Rightarrow x=\frac{-1 \pm \sqrt{3} i}{2} $
$\Rightarrow x=\omega, \omega^{2}$
Let $\alpha=\omega$ and $\beta=\omega^{2}$
$\Rightarrow \alpha^{58}=\omega^{58}=\omega^{57} \cdot \omega=\omega=\alpha$
and $\beta^{19}=\left(\omega^{2}\right)^{19}=\omega^{38}=\omega^{36} \cdot \omega^{2}=\omega^{2}=\beta$
$\Rightarrow$ the quadratic equation is $x^{2}+x+1=0$
$\Rightarrow a=1, b=1, c=1$
$\Rightarrow a+b+c=3$