α and β are the roots of quadratic equation x2−x+1=0 x2−x+1=0 x=
1±√1−4
2
=
1±√−3
2
=
1±√3i
2
=−ω and −ω2 [where, w is cube roots of unity] ∴−ω−ω2 are roots of quadratic equation x2−x+1=0 Let α=−w, then β=−ω2 Now, α2009+β2009