For which value of ' $k$ ', the roots of equation $2 x^2+5 x+k=0$ are rational?
For which value of ' $k$ ', the roots of equation $2 x^2+5 x+k=0$ are rational?
$\frac{5}{8}$
$\frac{25}{8}$
$\frac{25}{4}$
$\frac{5}{4}$
Solution
Given, quadratic equation is
$
2 x^2+5 x+k=0
$
Quadratic equation have rational roots when $D$ is perfect square,
$
\begin{aligned}
& D=b^2-4 a c=25-4 \cdot 2 k \\
& D=25-8 k
\end{aligned}
$
from options checking $k=\frac{25}{8}$ makes $D$ is perfect square
$
\therefore \quad k=\frac{25}{8}
$
Hence, option (2) is correct