5/(3^х-6)+5/(3^х+6)=2

To solve the equation

\[ \frac{5}{3^x - 6} + \frac{5}{3^x + 6} = 2, \]

let's first simplify the left-hand side.

We can combine the fractions:

\[ \frac{5(3^x + 6) + 5(3^x - 6)}{(3^x - 6)(3^x + 6)} = \frac{5(3^x + 6 + 3^x - 6)}{(3^x - 6)(3^x + 6)}. \]

Simplifying the numerator:

\[ 5(3^x + 6 + 3^x - 6) = 5(2 \cdot 3^x) = 10 \cdot 3^x. \]

So, we have

\[ \frac{10 \cdot 3^x}{(3^x - 6)(3^x + 6)} = 2. \]

Now, cross-multiplying gives:

\[ 10 \cdot 3^x = 2(3^x - 6)(3^x + 6). \]

Expanding the right-hand side:

\[ 2(3^x - 6)(3^x + 6) = 2((3^x)^2 - 36) = 2(3^{2x} - 36). \]

So, we have:

\[ 10 \cdot 3^x = 2 \cdot 3^{2x} - 72. \]

Rearranging gives:

\[ 2 \cdot 3^{2x} - 10 \cdot 3^x - 72 = 0. \]

Now, let \( y = 3^x \). Then the equation simplifies to:

\[ 2y^2 - 10y - 72 = 0. \]

Dividing the entire equation by 2 gives:

\[ y^2 - 5y - 36 = 0. \]

Now, we can solve this quadratic equation using the quadratic formula:

\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-36)}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 + 144}}{2}. \]

Calculating \( \sqrt{169} \):

\[ y = \frac{5 \pm 13}{2}. \]

Thus, we have two solutions for \( y \):

1. \( y = \frac{18}{2} = 9 \)
2. \( y = \frac{-8}{2} = -4 \) (not valid since \( y = 3^x > 0 \))

So the only valid solution is \( y = 9 \).

Now recall that \( y = 3^x \), so:

\[ 3^x = 9 \Rightarrow 3^x = 3^2. \]

Then, equating the exponents:

\[ x = 2. \]

Thus, the solution to the original equation is:

\[ \boxed{2}. \]