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): \( \quad \quad x^2 (p^3 - q) - x(5p^3 + 2q) + (p^3 - q) = 0 \).

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