Question

If $\alpha and\beta$ are the roots of the equation ax² + bx + c = 0. Find the value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }.$

Answer 1

$$\begin{gathered} \mathrm{To}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{\alpha }},\mathrm{first}\mathrm{we}\mathrm{need}\mathrm{to}\mathrm{express}\mathrm{it}\mathrm{in}\mathrm{terms}\mathrm{of}(\mathrm{\alpha }+\mathrm{\beta })\mathrm{and}\mathrm{\alpha \beta }. \\ \alpha +\beta =\frac{-b}{a}\text{and}\alpha \beta =\frac{c}{a} \\ \mathrm{Thus}, \\ \frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta } \\ =\frac{{\alpha }^2+{\beta }^2+2\alpha \beta -2\alpha \beta }{\alpha \beta } \\ =\frac{(\alpha +\beta {)}^2-2\alpha \beta }{\alpha \beta } \\ \mathrm{On}\mathrm{substituting}\mathrm{\alpha }+\mathrm{\beta }=\frac{-b}{a}and\mathrm{\alpha \beta }=\frac{c}{a},\mathrm{we}\mathrm{get}: \\ \frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{\alpha }}=\frac{{\left({\displaystyle \frac{-b}{a}}\right)}^2-2\left({\displaystyle \frac{c}{a}}\right)}{{\displaystyle \frac{c}{a}}} \\ =\frac{{\displaystyle \frac{b^2}{a^2}}-{\displaystyle \frac{2c}{a}}}{{\displaystyle \frac{c}{a}}} \\ =\frac{b^2-2ca}{ac} \end{gathered}$$