Let p and q be real numbers such that p≠0, p3≠q, and p3≠−q. If α and β are nonzero complex numbers satisfying α+β=−p and α3+β3=q, then a quadratic equation having α/β and β/α as its roots is

α3+β3=q⇒ (α+β)3−3αβ(α+β)=q⇒ −p3+3pαβ=q⇒αβ=q+p33pRequired equation isx2−αβ+βαx+αβ⋅βα=0x2−(α2+β2)αβx+1=0⇒ x2−(α+β)2−2αβαβx+1=0⇒x2−p2−2p3+q3pp3+q3px+1=0⇒(p3+q)x2−(3p3−2p3−2q)x+(p3+q)=0⇒ (p3+q)x2−(p3−2q)x+(p3+q)=0