Mathematics - Quadratic Equation Question with Solution | TestHub

The equation $x^2-px+r=0$ has roots $(\alpha ,\beta )$ and the equation $x^2-qx+r=0$ has roots $$\begin{array}{ll}\left(\frac{\alpha }{2},2\beta \right).& \\ \Rightarrow & r=\alpha \beta \text{ and }\alpha +\beta =p\text{ and }\frac{\alpha }{2}+2\beta =q\\ \Rightarrow & \beta =\frac{2q-p}{3}\text{ and }\alpha =\frac{2(2p-q)}{3}\\ \Rightarrow & \alpha \beta =r=\frac{2}{9}(2q-p)(2p-q)\end{array}$$