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  4. If a= cos (2 π/7)+i sin (2 π/7) then the quadratic equation whose roots are α=a+a2+a4 and β=a3+a5+ a6 is

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Q. If $a=\cos \frac{2 \pi}{7}+i \sin \frac{2 \pi}{7}$ then the quadratic equation whose roots are $\alpha=a+a^2+a^4$ and $\beta=a^3+a^5+$ $a^6$ is

Complex Numbers and Quadratic Equations

Solution:

Sum of roots $=a+a^2+a^3+a^4+a^5+a^6=-1$
product of root $=3 a^7+\left(a+a^2+a^3+a^4+a^5+a^6\right)=3-1=2 $
$\Rightarrow$ quadratic equation is $x^2+x+2=0$