If $\mathrm{a}=\cos \left(\frac{2\mathrm{\pi }}{7}\right)+\mathrm{isin}\left(\frac{2\mathrm{\pi }}{7}\right)$ where $\mathrm{i}=\sqrt{-1}$, then quadratic equation whose roots are $\mathrm{\alpha }=\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^4\mathrm{and}\mathrm{\beta }={\mathrm{a}}^3+{\mathrm{a}}^5+{\mathrm{a}}^6$

$\mathrm{a}=\cos \left(\frac{2\mathrm{\pi }}{7}\right)+\mathrm{isin}\left(\frac{2\mathrm{\pi }}{7}\right)$ is 7 th root of  unity

a⁷=1,a+a²+a³+a⁴+a⁵+a⁶=0

$\mathrm{\alpha }=\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^4\mathrm{and}\mathrm{\beta }={\mathrm{a}}^3+{\mathrm{a}}^5+{\mathrm{a}}^6$

$\alpha +\beta =-1,\alpha \beta =\left(a+a^2+a^4\right)\left(a^3+a^5+a^6\right)=2$

quadratic equation whose roots are $\mathrm{\alpha }=\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^4\mathrm{and}\mathrm{\beta }={\mathrm{a}}^3+{\mathrm{a}}^5+{\mathrm{a}}^6$

is $$\begin{gathered} x^2-x\left(\alpha +\beta \right)+\alpha \beta =0 \\ \Rightarrow x^2+x+2=0 \end{gathered}$$