To solve the quadratic equation \( 3(x - 4)^2 = 75 \), let's first isolate the quadratic expression.
- Divide both sides by 3:
\[ (x - 4)^2 = 25 \]
- Next, take the square root of both sides:
\[ x - 4 = \pm 5 \]
This gives us two equations to solve:
Case 1: \[ x - 4 = 5 \] Add 4 to both sides: \[ x = 9 \]
Case 2: \[ x - 4 = -5 \] Add 4 to both sides: \[ x = -1 \]
Thus, the solutions to the equation are:
\[ x = -1 \quad \text{and} \quad x = 9 \]
Therefore, the correct answer is:
\( x = -1 \) and \( x = 9 \).