Mathematics - Quadratic Equation Question with Solution | TestHub

We have $\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}$\ Let $\gamma ,\delta$ be the roots of ${\mathrm{a}}^3{\mathrm{x}}^2+\mathrm{abcx}+{\mathrm{c}}^3=0$.

Then $\gamma ,\delta =\frac{-abc\pm \sqrt{(abc{)}^2-4a^3{c}^3}}{2a^3}=\frac{ac\left\{-b\pm \sqrt{b^2-4ac}\right\}}{2a^3}=\frac{c}{2a}\left\{-\frac{b}{a}\pm \sqrt{{\left(\frac{b}{a}\right)}^2-4\frac{c}{a}}\right\}$

$\begin{array}{rlr}& =\frac{1}{2}(\alpha \beta )\left\{(\alpha +\beta )\pm \sqrt{(\alpha +\beta {)}^2-4\alpha \beta }\right\}& \end{array}$

$=\frac{1}{2}(\alpha \beta )\{(\alpha +\beta )\pm (\alpha -\beta )\}={\alpha }^2\beta ,\alpha {\beta }^2$

Thus, roots of $a^3{x}^2+abcx+c^3=0$ are ${\alpha }^2\beta$ and $\alpha {\beta }^2$