Solve. \( \sqrt{x-9}+\sqrt{x+6}=5 \)

To solve the equation \( \sqrt{x-9} + \sqrt{x+6} = 5 \), we can start by isolating one of the square root terms. Let's isolate \( \sqrt{x-9} \): \[ \sqrt{x-9} = 5 - \sqrt{x+6} \] Next, square both sides to eliminate the square root: \[ x - 9 = (5 - \sqrt{x+6})^2 \] Expanding the right side gives: \[ x - 9 = 25 - 10\sqrt{x+6} + (x + 6) \] This simplifies to: \[ x - 9 = x + 31 - 10\sqrt{x+6} \] Subtracting \( x \) from both sides yields: \[ -9 = 31 - 10\sqrt{x+6} \] Now, rearranging gives: \[ 10\sqrt{x+6} = 31 + 9 \] So, \[ 10\sqrt{x+6} = 40 \] Dividing both sides by 10 results in: \[ \sqrt{x+6} = 4 \] Now square both sides again: \[ x + 6 = 16 \] Subtracting 6 from both sides we find: \[ x = 10 \] To ensure that this solution is valid, let's check it in the original equation: \[ \sqrt{10-9} + \sqrt{10+6} = \sqrt{1} + \sqrt{16} = 1 + 4 = 5 \] Since both sides of the equation match, the solution is indeed valid. Thus, the solution to the equation is \[ \boxed{10} \]