Question

for what values of p the quadratic equation 2x^2​-5x+p =0 one roots equal to twice the other roots

Answer 1

$$\begin{gathered} 2x^2-5x+p=0 \\ D={\left(-5\right)}^2-4\left(2\right)p \\ =25-8p \\ x=\frac{5\pm \sqrt{25-8p}}{4} \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ \frac{5+\sqrt{25-8p}}{4}=2\left(\frac{5-\sqrt{25-8p}}{4}\right) \\ \Rightarrow 5+\sqrt{25-8p}=10-2\sqrt{25-8p} \\ \Rightarrow 3\sqrt{25-8p}=5 \\ \Rightarrow 9\left(25-8p\right)=25 \\ \Rightarrow 225-72p=25 \\ \Rightarrow -72p=-200 \\ \Rightarrow p=\frac{200}{72}=\frac{25}{9} \end{gathered}$$