1. Given equation

We need to find the number of solutions of

Since $\sqrt{x}$ appears in the denominator, the domain is

Because the product is zero, we solve:

2. Substitute

Let

Since $x>0$, we have $t>0$. Also,

So the equation becomes

Thus we solve the two quadratics.

3. First factor

Factorizing,

So,

Now convert back using $t=\frac{1}{\sqrt{x}}$:

• If $t=\frac13$, then $$\sqrt{x}=3 \implies x=9.$$
• If $t=\frac23$, then $$\sqrt{x}=\frac32 \implies x=\frac94.$$

So this factor gives 2 solutions.

4. Second factor

Factorizing,

So,

Convert back:

• If $t=\frac12$, then $$\sqrt{x}=2 \implies x=4.$$
• If $t=3$, then $$\sqrt{x}=\frac13 \implies x=\frac19.$$

So this factor gives 2 more solutions.

5. Check for repetition

The four values obtained are

All are distinct and all satisfy $x>0$.

Hence total number of solutions is

6. Option matching

Option $\boxed{D}$ corresponds to $4$.

So the stored correct answer is correct.