∴ a7=cos2π+isin2π [∵eiθ=cosθ+isinθ] = 1 Also, α=a+a2+a4,β=a3+a5+a6 then the sum of roots, S=α+β=a+a2+a3+a4+a5+a6 ⇒ S=
a(1−a6)
1−a
=
a−a7
1−a
=
a−1
1−a
=−1[∵a7=1] Product of the roots, P=αβ=(a+a2+a4)(a3+a5+a6) =a4+a5+1+a6+1+a2+1+a+a3[∵a7=1] =3+(a+a2+a3+a4+a5+a6)=3−1=2 Hence, the required quadratic equation is x2+x+2=0