We are given the equation:

$\frac{\sqrt{a + x} + \sqrt{a - x}}{\sqrt{a + x} - \sqrt{a - x}} = \frac{a}{x}$

We will start by simplifying the left-hand side (LHS) using the difference of squares method.

Step 1: Multiply the numerator and denominator by the conjugate of the denominator.

The conjugate of $\sqrt{a + x} - \sqrt{a - x}$ is $\sqrt{a + x} + \sqrt{a - x}$. Therefore, multiply both the numerator and the denominator by $\sqrt{a + x} + \sqrt{a - x}$:

$\frac{\left( \sqrt{a + x} + \sqrt{a - x} \right) \cdot \left( \sqrt{a + x} + \sqrt{a - x} \right)}{\left( \sqrt{a + x} - \sqrt{a - x} \right) \cdot \left( \sqrt{a + x} + \sqrt{a - x} \right)}$

Step 2: Simplify the denominator using the difference of squares formula.

The denominator becomes:

$(\sqrt{a + x})^2 - (\sqrt{a - x})^2 = (a + x) - (a - x) = 2x$

Step 3: Simplify the numerator.

The numerator is:

$\left( \sqrt{a + x} + \sqrt{a - x} \right)^2 = (\sqrt{a + x})^2 + 2\sqrt{(a + x)(a - x)} + (\sqrt{a - x})^2$ $= (a + x) + 2\sqrt{(a + x)(a - x)} + (a - x)$ $= a + x + a - x + 2\sqrt{(a + x)(a - x)}$ $= 2a + 2\sqrt{(a + x)(a - x)}$

Step 4: Substitute the simplified numerator and denominator back into the expression.

We now have:

$\frac{2a + 2\sqrt{(a + x)(a - x)}}{2x}$

Step 5: Cancel out the common factor of 2.

The expression simplifies to:

$\frac{a + \sqrt{(a + x)(a - x)}}{x}$

Step 6: Simplify the square root term.

Using the difference of squares, we simplify $\sqrt{(a + x)(a - x)}$:

$\sqrt{(a + x)(a - x)} = \sqrt{a^2 - x^2}$

Step 7: Final expression.

The left-hand side becomes:

$\frac{a + \sqrt{a^2 - x^2}}{x}$

Since this is already in a simplified form, we compare it with the right-hand side, $\frac{a}{x}$.

For the equation to hold true, the square root term must be 0, i.e., $\sqrt{a^2 - x^2} = 0$.

Thus, $a^2 - x^2 = 0$, which gives $a^2 = x^2$, or $a = x$ or $a = -x$.

Hence, the equation holds when $a = x$ or $a = -x$.

---

Would you like further details or have any questions on the steps?

Here are some related questions you might explore:

1. What are the general properties of square roots in equations?
2. How can we manipulate expressions involving radicals?
3. Why is multiplying by a conjugate helpful in simplifying expressions?
4. What are other uses of the difference of squares technique?
5. How can we solve equations involving square roots in general?

Tip: Always check if the final solution satisfies the original equation by substituting back the values.