If p and q are non-zero real numbers, ^3+^3=-p, =q, then a quadratic equation whose roots …

Question

If $p$ and $q$ are non-zero real numbers, $\alpha^3+\beta^3=-p$, $\alpha\beta=q$, then a quadratic equation whose roots are $\dfrac{\alpha^2}\beta,\dfrac{\beta^2}\alpha$ is:

Accepted Answer

$$\frac{\alpha^2}\beta+\frac{\beta^2}\alpha=\frac{\alpha^3+\beta^3}{\alpha\beta}=\frac{-p}q,\qquad \frac{\alpha^2}\beta\cdot\frac{\beta^2}\alpha=\alpha\beta=q.$$

Required equation: $x^2+\dfrac pqx+q=0 \implies qx^2+px+q^2=0.$

Solutions · 1