Solve the equation. \( \sqrt{9 x+117}=2 x+17 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The solution set is \( \{ \). (Use a comma to separate answers as needed.) B. There is no solution.

To solve the equation \( \sqrt{9x + 117} = 2x + 17 \), first square both sides to eliminate the square root: \[ 9x + 117 = (2x + 17)^2 \] Expanding the right side gives: \[ 9x + 117 = 4x^2 + 68x + 289 \] Now, rearranging the equation yields: \[ 0 = 4x^2 + 68x + 289 - 9x - 117 \] \[ 0 = 4x^2 + 59x + 172 \] Next, apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 4 \), \( b = 59 \), and \( c = 172 \). The discriminant is: \[ b^2 - 4ac = 59^2 - 4 \cdot 4 \cdot 172 = 3481 - 2752 = 728 \] Since the discriminant is positive, we can find real solutions. Calculating: \[ x = \frac{-59 \pm \sqrt{728}}{8} = \frac{-59 \pm 27}{8} \] This results in: \[ x_1 = \frac{-32}{8} = -4, \quad x_2 = \frac{-86}{8} = -10.75 \] Now we need to check both solutions in the original equation: For \( x = -4 \): \[ \sqrt{9(-4) + 117} = \sqrt{-36 + 117} = \sqrt{81} = 9 \] \[ 2(-4) + 17 = -8 + 17 = 9 \quad \text{(valid)} \] For \( x = -10.75 \): Calculating \( 9(-10.75) + 117 \), we find: \[ 9 \cdot -10.75 + 117 = -96.75 + 117 = 20.25 \Rightarrow \sqrt{20.25} \neq -10.75 \] This solution does not hold. Thus, the only valid solution is \( x = -4 \). The solution set is \( \{-4\} \).