To solve the equation
[tex]\[
\frac{2}{\sqrt{x} + 2} + \frac{\sqrt{x}}{\sqrt{x} - 2} = \frac{x + 4}{x - 4},
\][/tex]
we will follow a step-by-step approach:
1. Identify the Domain:
First, consider the domain of the equation. The expressions [tex]\(\frac{2}{\sqrt{x} + 2}\)[/tex], [tex]\(\frac{\sqrt{x}}{\sqrt{x} - 2}\)[/tex], and [tex]\(\frac{x + 4}{x - 4}\)[/tex] are only defined for [tex]\(x \ge 0\)[/tex] (since [tex]\(\sqrt{x}\)[/tex] is only defined for non-negative [tex]\(x\)[/tex]) and [tex]\(x \not= 4\)[/tex] (to avoid division by zero on the right-hand side).
2. Simplify the Equation:
Consider a common denominator to combine the fractions on the left-hand side. The common denominator for [tex]\(\frac{2}{\sqrt{x} + 2}\)[/tex] and [tex]\(\frac{\sqrt{x}}{\sqrt{x} - 2}\)[/tex] is [tex]\((\sqrt{x} + 2)(\sqrt{x} - 2)\)[/tex].
[tex]\[
\frac{2(\sqrt{x} - 2) + \sqrt{x}(\sqrt{x} + 2)}{(\sqrt{x} + 2)(\sqrt{x} - 2)} = \frac{x + 4}{x - 4}
\][/tex]
3. Combine the Numerators:
Distribute and combine the numerators of the fractions on the left-hand side:
[tex]\[
\frac{2\sqrt{x} - 4 + x + 2\sqrt{x}}{x - 4} = \frac{x + 4}{x - 4}
\][/tex]
[tex]\[
\frac{x + 4\sqrt{x} - 4}{x - 4}
\][/tex]
4. Equate the Numerators:
Since the denominators are equal, the numerators must be equal:
[tex]\[
x + 4\sqrt{x} - 4 = x + 4
\][/tex]
5. Simplify the Equation:
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
4\sqrt{x} - 4 = 4
\][/tex]
Add 4 to both sides:
[tex]\[
4\sqrt{x} = 8
\][/tex]
Divide both sides by 4:
[tex]\[
\sqrt{x} = 2
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
Square both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 4
\][/tex]
7. Verify the Solution:
Substitute [tex]\(x = 4\)[/tex] back into the original equation to verify. However, recall our domain restrictions: [tex]\(x \not= 4\)[/tex]. This value [tex]\(x = 4\)[/tex] makes the denominator of the right-hand side zero, which is undefined.
Thus, there are no valid solutions within the given domain. Hence, the solution set is the empty set:
[tex]\[
\boxed{\emptyset}
\][/tex]
