Mathematics - Quadratic Equation Question with Solution | TestHub

If $\alpha ,\beta ,\gamma$ are the roots of the equation ${\mathrm{x}}^3+3\mathrm{x}+\mathrm{c}=0$ then find the value of $\frac{1}{27}\sum (\alpha -\beta {)}^2(\beta -\gamma {)}^2$.

$$\begin{array}{rlrlrl}& {\partial }_2=(\beta -\gamma )(\gamma -\alpha )& {\partial }_3=(\gamma -\alpha )(\alpha -\beta )& \text{ required part is }\frac{{\partial }_1^2+{\partial }_2^2+{\partial }_3^2}{27}& {\partial }_1{\partial }_2+{\partial }_2{\partial }_3+{\partial }_2{\partial }_1=0\&{\partial }_1+{\partial }_2+{\partial }_3=9& \frac{{\partial }_1^2+{\partial }_2^2+{\partial }_3^2}{27}=\frac{81}{27}=3\end{array}$$