Thank you for reporting, we will resolve it shortly
Q.
The common roots of the equations $x^{12}-1=0$, $x^{4}+x^{2}+1=0$ are
Complex Numbers and Quadratic Equations
Solution:
$x^{12}-1=\left(x^{6}+1\right)\left(x^{6}-1\right)$
$=\left(x^{6}+1\right)\left(x^{2}-1\right)\left(x^{4}+x^{2}+1\right)$
Common roots are given by $x^{4}+x^{2}+1=0$
$\therefore x^{2}=\frac{-1 \pm i \sqrt{3}}{2}=\omega, \omega^{2}$
or $\omega^{4}, \omega^{2}$
$ \left(\because \omega^{3}=1\right)$
or $x=\pm \omega^{2}, \pm \omega$