Question

If one root of the quation $a{x}^2+bx+c=0$ is double the other, then the relation between $a,b,c$, is

• A. $2b^2=9ac$
• B. $b^2=ac$
• C. $b^2=3ac$
• D. $b^2=9ac$

Answer 1

The correct option is A $2b^2=9ac$

If the roots of the equation $a{x}^2+bx+c=0$ are $\alpha$ and $\beta$

Then $\alpha +\beta =\frac{-b}{a}$ ....(1)

and $\alpha \beta =-\frac{c}{a}$ .....(2)

Let $\beta =2\alpha$

Using (1) and (2), we have

$\alpha +2\alpha =\frac{-b}{a}$

$\Rightarrow \alpha =\frac{-b}{3a}$ .....(3)

and $2{\alpha }^2=\frac{c}{a}$ .....(4)

Substituting (3) in (4), we get

$2{(-\frac{b}{3a})}^2=\frac{c}{a}$

$\Rightarrow 2b^2=9ca$