Here's the simplified version of the equation for $a \neq 0$ and $x \neq 0$ :

$$x^2 + ab = 0$$

To simplify the expression $\frac{x}{a} + \frac{b}{x} = 0$, let's follow these steps:

1. First, let's find a common denominator for the fractions. The common denominator for $a$ and $x$ is $ax$.

   $$\frac{x}{a} + \frac{b}{x} = 0$$

   Rewrite each fraction with the common denominator:

   $$\frac{x^2}{ax} + \frac{ab}{ax} = 0$$

2. Combine the fractions over the common denominator:

   $$\frac{x^2 + ab}{ax} = 0$$

3. For the fraction to be zero, the numerator must be zero (since the denominator $ax$ is not zero for $a \neq 0$ and $x \neq 0$ ).

   $$x^2 + ab = 0$$

4. Solve for $x$:

   $$x^2 = -ab$$

   Taking the square root of both sides:

   $$x = \pm \sqrt{-ab}$$

5. Note that $\sqrt{-ab}$ is a complex number if $ab$ is positive, since the square root of a negative number involves the imaginary unit $i$.

Thus, the solution to the given equation is:

$$x = \pm \sqrt{-ab}$$

If $ab$ is positive, then the solutions can be written in terms of imaginary numbers:

$$x = \pm i\sqrt{ab}$$