If $\mathrm{a}=\cos \frac{2\mathrm{\pi }}{7}+\mathrm{i} \sin \frac{2\mathrm{\pi }}{7}, \mathrm{\alpha }=\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^4$ and $\mathrm{\beta }={\mathrm{a}}^3+{\mathrm{a}}^5+{\mathrm{a}}^6$ then $\mathrm{\alpha }, \mathrm{\beta }$ are the roots of the equation

$\begin{array}{l}Ifa=\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7}then\\ aisrootof{x}^7-1=0\Rightarrow a^7=1\\ \alpha =a+a^2+a^4,\beta =a^3+a^5+a^6\\ \alpha +\beta =a+a^2+a^3+a^4+a^5+a^6\\ =-1[1+a+a^2+a^3+a^4+a^5+a^6]=0\\ \alpha \beta =(a+a^2+a^4)(a^3+a^5+a^6)\\ =a^4+a^6+a^7+a^5+a^7+a^8+a^7+a^9+a^{10}\\ =3\left(a^7\right)+a^7.a^1+a^7.a^2+a^7.a^3+a^4+a^5+a^6\\ =3+a+a^2+a^3+a^4+a^5+a^6\\ =3-1\\ =2\\ \mathrm{Re}quiredEquation{x}^2-(\alpha +\beta )x+\alpha \beta =0\\ \Rightarrow x^2+x+2=0\end{array}$