8. Solve the equation $\sqrt{15 x^{2}-52 x+45} \cdot(3-\sqrt{5 x-9}-\sqrt{3 x-5})=1$.
Solution. Rewrite our equation in the form

$$
\sqrt{3 x-5} \cdot \sqrt{5 x-9} \cdot(3-\sqrt{5 x-9}-\sqrt{3 x-5})=1
$$

Such a transformation is possible because the solution to the original equation exists only for $x>\frac{9}{5}$. Let $\sqrt{3 x-5}=a>0, \sqrt{5 x-9}=b>0$. We have

$$
a+b+\frac{1}{a b}=3
$$

Apply the Cauchy inequality to the left side

$$
a+b+\frac{1}{a b} \geq 3 \sqrt[3]{a \cdot b \cdot \frac{1}{a b}}=3
$$

Thus, for the solution of our equation, it is necessary that the Cauchy inequality holds as an equality and $a=b=\frac{1}{a b}$.

Answer: 2.