Mathematics - Quadratic Equation Question with Solution | TestHub

Let $\alpha$ and $\beta$ be the roots of the equation, $81x^2+kx+256=0$ Given $(\alpha {)}^{\frac{1}{3}}=\beta \Rightarrow \alpha ={\beta }^3$ $\because$ Product of the roots $=\frac{256}{81}$ $\therefore (\alpha )(\beta )=\frac{256}{81}$ $\Rightarrow {\beta }^4={\left(\frac{4}{3}\right)}^4\Rightarrow \beta =\frac{4}{3}\Rightarrow \alpha =\frac{64}{27}$ $\because$ Sum of the roots $=-\frac{k}{81}$ $\therefore \alpha +\beta =-\frac{k}{81}\Rightarrow \frac{4}{3}+\frac{64}{27}=-\frac{k}{81}$ $\Rightarrow k=-300$