Complete the square to solve the following quadratic equation: x^2 - 22 + 1 = 15.

Question

A. x = -11 + 3 square root of 15, -11 - 3 square root of 15
B. x = 11 + 3 square root of 15, 11 - 3 square root of 15
C. x = -11 + square root of 14, -11 - square root of 14
D. x = 11 + square root of 14, 11 - square root of 14

Answer 1

To solve the quadratic equation x^2 - 22x + 1 = 15, we first move all terms to one side of the equation:
x^2 - 22x + 1 - 15 = 0.

Next, we simplify the left side of the equation:
x^2 - 22x - 14 = 0.

To complete the square, we take half of the coefficient of x (-22/2 = -11) and square it (-11)^2 = 121. We add this value to both sides of the equation:
x^2 - 22x + 121 - 14 + 121 = 121,
(x - 11)^2 + 107 = 121.

Now, we subtract 107 from both sides of the equation:
(x - 11)^2 = 121 - 107,
(x - 11)^2 = 14.

Taking the square root of both sides, with consideration for both positive and negative square root:
x - 11 = ±√14.

Finally, we add 11 to both sides of the equation, giving us the solutions for x:
x = 11 ± √14.

Therefore, the correct answer is D. x = 11 + √14, 11 - √14.