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: \[ \boxed{2}. \]