Mathematics - Quadratic Equation Question with Solution | TestHub

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 $$\begin{array}{rl}& x^2-\left(\frac{-p}{q}\right)x+q=0\\ & \Rightarrow x^2+\frac{p}{q}x+q=0\Rightarrow q{x}^2+px+q^2=0\end{array}$$