# An algebra problem by Oleg Silkin

Algebra Level pending

Let $$p$$ and $$q$$ be real numbers such that $$p\ne0, p^3\ne q, p^3 \ne -q$$. If $$\alpha$$ and $$\beta$$ are non-zero complex numbers satisfying $$\alpha + \beta = -p$$ and $$\alpha^3 + \beta^3 = q$$, then a quadratic equation having $$\dfrac{ \alpha}{\beta} + \dfrac{\beta}{\alpha}$$ as its roots is

(A): $$\quad \quad x^2 (p^3 + q) - x(p^3 + 2q) + (p^3 + q) = 0$$.
(B): $$\quad \quad x^2 (p^3 + q) - x(p^3 - 2q) + (p^3 + q) = 0$$.
(C): $$\quad \quad x^2 (p^3 - q) - x(5p^3 - 2q) + (p^3 - q) = 0$$.
(D): $$\quad \quad x^2 (p^3 - q) - x(5p^3 + 2q) + (p^3 - q) = 0$$.

×