Mathematics - Quadratic Equation Question with Solution | TestHub

Let $\alpha ,\beta ,\gamma ,\delta$ be four positive real roots of the given equation. Then:

$$\begin{array}{rl}\alpha +\beta +\gamma +\delta & =-\mathrm{p}\\ \Sigma \alpha \beta & =\mathrm{q}\\ \Sigma \alpha \beta \gamma & =-\mathrm{r}\\ \alpha \beta \gamma \delta & =5\end{array}$$

Using A.M. $\ge$ G.M.:

$$\begin{array}{rl}\frac{\alpha +\beta +\gamma +\delta }{4}& \ge (\alpha \beta \gamma \delta {)}^{1/4}\\ \frac{\Sigma \alpha \beta \gamma }{4}& \ge ({\alpha }^3{\beta }^3{\gamma }^3{\delta }^3{)}^{1/4}\\ \frac{(\Sigma \alpha )\cdot (\Sigma \alpha \beta \gamma )}{16}& \ge (\alpha \beta \gamma \delta )\\ \mathrm{pr}& \ge 80\end{array}$$