6.039. $\left(\frac{x+5}{x}\right)^{\frac{1}{2}}+4\left(\frac{x}{x+5}\right)^{\frac{1}{2}}=4$.
## Solution.

From the condition we have

$$
\sqrt{\frac{x+5}{x}}+4 \sqrt{\frac{x}{x+5}}-4=0 \Leftrightarrow \sqrt{\frac{x+5}{x}}+\frac{4}{\sqrt{\frac{x}{x+5}}}-4=0
$$

Let

$$
\begin{aligned}
& \sqrt{\frac{x+5}{x}}=y > 0: y+\frac{4}{y}-4=0 \Leftrightarrow y^{2}-4 y+4=0 \Leftrightarrow \\
& \Leftrightarrow(y-2)^{2}=0 \Leftrightarrow y-2=0 \Leftrightarrow y=2
\end{aligned}
$$

Then $\sqrt{\frac{x+5}{x}}=2$. By verification, we confirm that this expression satisfies the condition.

From this, $\frac{x+5}{x}=4 \Leftrightarrow x+5=4 x, x=\frac{5}{3}$.

Answer: $x=\frac{5}{3}$.