Final answer:
To complete the square, Samantha needs to add the square of half the coefficient of x to both sides of the equation. She can then solve for x by taking the square root of both sides.
Step-by-step explanation:
To complete the square, Samantha needs to add the square of half the coefficient of x to both sides of the equation. In this case, the coefficient of x is -12, so half of that is -6. So, she needs to add (-6)^2 = 36 to both sides.
The equation then becomes x^2 - 12x + 36 = 40 + 36, which simplifies to (x - 6)^2 = 76.
To solve for x, Samantha needs to take the square root of both sides of the equation. So, x - 6 = ±√76. Solving for x gives us two possible answers: x = 6 + √76 and x = 6 - √76.