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 $a{x}^2+bx+c=0$, then find the value of $a+b+c$.

Answer 1

Answer: 3

$x^2+x+1=0$
$\Rightarrow x=\frac{-1\pm \sqrt{3}i}{2}$
$\Rightarrow x=\omega ,{\omega }^2$
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 $x^2+x+1=0$
$\Rightarrow a=1,b=1,c=1$
$\Rightarrow a+b+c=3$