Mathematics - Quadratic Equation Question with Solution | TestHub

Let $\alpha$ and $\beta$ be the roots of $x^2-5x+3=0$. By Vieta's formulas, $\alpha +\beta =5$ and $\alpha \beta =3$. We need to find $\frac{{\alpha }^2}{\beta }+\frac{{\beta }^2}{\alpha }=\frac{{\alpha }^3+{\beta }^3}{\alpha \beta }$. Using the identity ${\alpha }^3+{\beta }^3=(\alpha +\beta )((\alpha +\beta {)}^2-3\alpha \beta )$, we substitute the values: ${\alpha }^3+{\beta }^3=(5)((5{)}^2-3(3))=5(25-9)=5(16)=80$. Therefore, the expression is $\frac{80}{3}$.