What are the solutions to the quadratic equation 3(x − 4)2 = 75?

x = −9 and x = 1
x = −5 and x = 5
x = −4 and x = 4
x = −1 and x = 9

1 answer

To solve the quadratic equation \( 3(x - 4)^2 = 75 \), let's first isolate the quadratic expression.

  1. Divide both sides by 3:

\[ (x - 4)^2 = 25 \]

  1. 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 \( x = 9 \).