$5^{2 x}-(\sqrt{5}+25) \cdot 5^{x}+25 \sqrt{5}=0$

Step 1 ：Substitute $y = 5^x$ and rewrite the equation in terms of $y$: $y^2 - (sqrt{5} + 25)y + 25sqrt{5} = 0$

Step 2 ：Solve the quadratic equation for $y$ using the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -(sqrt{5} + 25)$, and $c = 25sqrt{5}$

Step 3 ：Calculate the values of $y$: $y_1 = 25$ and $y_2 = 2.236067977499788$

Step 4 ：Find the corresponding values of $x$ by solving $5^x = y$ for each value of $y$: $x = \log_5{y}$

Step 5 ：Calculate the values of $x$: $x_1 = 2$ and $x_2 = 0.5$

Step 6 ：\(\boxed{x = 2, x = 0.5}\)