Question

If $\alpha ,\beta$ are the roots of $x^2-2x+4=0$, Find the value of ${\alpha }^6+{\beta }^6$.

Answer 1

$x^2-2x+4=0$
$\alpha ,\beta =\frac{2\pm \sqrt{4-16}}{2}=1\pm i\sqrt{3}=2cis(\pm \frac{\pi }{3})$
now, ${\alpha }^6+{\beta }^6=2^6(cis\left(\pi \right)+cis(-\pi ))=128$
Ans: 128