# Find the value of c for which the roots of 2x^2-21x+c=0 are in the ratio 1:2?

Jun 27, 2017

$c = 49$

#### Explanation:

When we have two roots of $a {x}^{2} + b x + c = 0$, then

sum of roots is $- \frac{b}{a}$ and product of roots ic $\frac{c}{a}$.

As roots of $2 {x}^{2} - 21 x + c = 0$ are in the ratio $1 : 2$, let the roots be $\alpha$ and $2 \alpha$.

Then $\alpha + 2 \alpha = \frac{21}{2}$ i.e. $3 \alpha = \frac{21}{2}$ or $\alpha = \frac{7}{2}$.

and hence as $\frac{c}{2}$ is product of roots, we have

$\frac{c}{2} = \frac{7}{2} \times 2 \times \frac{7}{2} = \frac{49}{2}$

or $c = 49$