If p and q are non-zero real numbers and α^3+β^3=-p, α β=q , then a...

Given $\alpha^3+\beta^3=-p and$\alpha \beta=q$Let$\frac{\alpha^2}{\beta}$and$\frac{\beta^2}{\alpha}$be the root of required quadratic equation. So,$\frac{\alpha^2}{\beta}+\frac{\beta^2}{\alpha}=\frac{\alpha^3+\beta^3}{\alpha \beta}=\frac{-p}{q}$and$\frac{\alpha^2}{\beta} \times \frac{\beta^2}{\alpha}=\alpha \beta=q$Hence, required quadratic equation is$$\begin{aligned}&x^2-\left(\frac{-p}{q}\right)x+q=0 \\ &\Rightarrow x²+\frac{p}{q}x+q=0 \Rightarrow q x²+p x+q²=0 \end{aligned}$$