Solve the quadratic equation. \( 3(x-4)^{2}+6=0 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution(s) is(are) \( x= \) (Simplify your answer. Type an exact answer, using radicals as needed. Use a comma to separate ans B. There are no real solutions.

To solve the equation \( 3(x-4)^{2}+6=0 \), we start by isolating the squared term. Subtract 6 from both sides: \[ 3(x-4)^{2} = -6 \] Next, divide both sides by 3: \[ (x-4)^{2} = -2 \] Since the square of a real number cannot be negative, we can conclude that there are no real solutions to this equation. Thus, the correct choice is: B. There are no real solutions. --- The concept of complex numbers can be a fun twist here! When you encounter a negative under a square root, it opens up the world of imaginary numbers. You would use \( i \) (where \( i = \sqrt{-1} \)) to express solutions in the complex plane. Quadratic equations with no real solutions often appear in physics and engineering, especially when dealing with scenarios with no valid physical occurrences, like finding the height of an object that can't reach a certain point due to gravity, leading to complex solutions that can reveal important insights!