Mathematics - Quadratic Equation Question with Solution | TestHub

Given $\alpha ,\beta$ are roots of $x^2-px+q=0$, so $\alpha +\beta =p$ and $\alpha \beta =q$. The sum of the new roots is $S^{\prime }=\frac{{\alpha }^2}{\beta }+\frac{{\beta }^2}{\alpha }=\frac{{\alpha }^3+{\beta }^3}{\alpha \beta }=\frac{(\alpha +\beta )((\alpha +\beta {)}^2-3\alpha \beta )}{\alpha \beta }=\frac{p(p^2-3q)}{q}$. The product of the new roots is $P^{\prime }=\frac{{\alpha }^2}{\beta }\cdot \frac{{\beta }^2}{\alpha }=\alpha \beta =q$. The required equation is $x^2-S^{\prime }x+P^{\prime }=0$, which is $x^2-\frac{p(p^2-3q)}{q}x+q=0$. Multiplying by $q$, we get $q{x}^2-p(p^2-3q)x+q^2=0$.