Question

Find the value of k for which one root of the quadratic equation kx² - 14x + 8 = 0 is six times the other

Answer 1

Let the first root be α.
then second root will be 6​α.
$$\begin{gathered} \mathrm{Sum}\mathrm{of}\mathrm{roots}=-\frac{b}{a} \\ \alpha +6\alpha =\frac{-\left(14\right)}{k}=\frac{14}{k} \\ 7\alpha =\frac{14}{k} \\ \alpha =\frac{2}{k} \\ \mathrm{and} \\ \mathrm{Product}\mathrm{of}\mathrm{roots}=\frac{c}{a} \\ \alpha \times 6\alpha =\frac{8}{k} \\ 6{\alpha }^2=\frac{8}{k} \\ 6{\left(\frac{2}{k}\right)}^2=\frac{8}{k} \\ k=3 \end{gathered}$$