Question

Find the condition that the roots of the equation $a{x}^2+bx+c=0$ be such that ,one root is n times the other

Answer 1

Let one root be $\alpha$ then the other root will be n$\alpha$ by given condition
Sum = $\alpha +n\alpha =-\frac{b}{a}or\alpha =\frac{b}{a(1+n)}...\left(1\right)$
Product = $n{\alpha }^2=\frac{c}{a}\therefore {\alpha }^2=\frac{c}{an}...\left(2\right)$
Now ${\alpha }^2=\left({\alpha }^2\right)\therefore \frac{c}{an}=\frac{b^2}{a^2(1+n{)}^2}by\left(1\right)and\left(2\right)$
$\therefore n{b}^2=ac(n+1{)}^2$