Question

If one root of quadratic equation $l{x}^2+mx+n=0$ is double the other, find the condition.

Answer 1

Given,

$\beta =2\alpha$ one root is the double of other root of the equation $l{x}^2+mx+n=0$

$x^2-(\alpha +2\alpha )x+\left(\alpha \right)\left(2\alpha \right)=0$

$x^2-3\alpha x+2{\alpha }^2=0$

comparing the equations, we get,

$l=1$

$m=-3\alpha \Rightarrow \alpha =-\frac{m}{3}$

$n=2{\alpha }^2$

$n=2{(-\frac{m}{3})}^2$

$\therefore n=\frac{2}{9}{m}^2$

Is the required condition.