For what value (s) of a is x + 1/4 x + a^2 a perfect square?

Answer: Option B

$$\begin{array}{rl}& x+\frac{1}{4}\sqrt{x}+a^2\\ & ={\left(\sqrt{x}\right)}^2+2\times \frac{1}{8}\times \sqrt{x}+a^2\\ & \left[\left({\text{A}}^2+\text{2AB}+{\text{B}}^2\right)={\left(\text{A}+\text{B}\right)}^2\right]\\ & \text{Here, A}=\sqrt{x}\text{ and }\\ & \text{B}=a\\ & \text{B}=\frac{1}{8}\\ & \therefore a=\frac{1}{8}\end{array}$$