Question

If $3:1$ is the ratio of the roots of the quadratic equation $2x^2+5x+m=0$, then the value of $m$ is equal to

• A. $\frac{1}{25}$
  
• B. $\frac{75}{32}$
  
• C. $\frac{81}{16}$
  
• D. $\frac{97}{16}$
  

Answer 1

The correct option is B $\frac{75}{32}$


Let $\alpha$ and $\beta$ be the roots of $2x^2+5x+m=0$
Given, $\alpha :\beta =3:1$
$\Rightarrow \frac{\alpha }{\beta }=\frac{3}{1}$
$\Rightarrow \alpha =3\beta$
Now, sum of roots, $\alpha +\beta =\frac{-5}{2}$
$\Rightarrow 3\beta +\beta =\frac{-5}{2}$
$\Rightarrow \beta =\frac{-5}{8}$
Also, product of roots, $\alpha \beta =\frac{m}{2}$
$\Rightarrow 3{\beta }^2=\frac{m}{2}$
$\Rightarrow 3\times {\left(\frac{-5}{8}\right)}^2=\frac{m}{2}$
$\Rightarrow m=3\times \frac{25}{64}\times 2=\frac{75}{32}$
Hence, the correct answer is option (2).