Question

Find the condition that one root of $a{x}^2+bx+c=0$ shall be $n$ times the other.

Answer 1

Given equation is $a{x}^2+bx+c=0$

Given that one root is $n$ time the other root

Let the roots be $p,np$

We have $p+np=-\frac{b}{a}$ and $n{p}^2=\frac{c}{a}$

$\Rightarrow p=-\frac{b}{a(1+n)}$

Now substitute this value of $p$ in $n{p}^2=\frac{c}{a}$

$\Rightarrow n\frac{b^2}{a^2(1+n{)}^2}=\frac{c}{a}$

$\Rightarrow n{b}^2=ac(1+n{)}^2$