6. Solve the equation $\sqrt{\frac{x+2}{x-3}}-6 \sqrt{\frac{x-3}{x+2}}+1=0$.
Solution: $\sqrt{\frac{x+2}{x-3}}-6 \sqrt{\frac{x-3}{x+2}}+1=0$, Domain of Definition $\frac{x+2}{x-3}>0, \frac{x+2}{x-3}=t>0$,

$\sqrt{t}-6 \frac{1}{\sqrt{t}}+1=0,(\sqrt{t})^{2}+\sqrt{t}-6=0,(\sqrt{t})_{1,2}=\frac{-1 \pm \sqrt{1+24}}{2}=\frac{-1 \pm 5}{2}=\left[\begin{array}{l}2 \\ -3-\text { not valid }\end{array}\right.$

$\frac{x+2}{x-3}=2^{2}, x+2=4 x-12, x=\frac{14}{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_8eeeb8d6ace591217257g-21.jpg?height=257&width=1468&top_left_y=143&top_left_x=137)

Answer: $x=\frac{14}{3}$.