Complete the square to solve the following quadratic equation: x^2−22x+1=15 .(1 point)

To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, we first rewrite the equation:

1. Move 15 to the left side:
\[
x^2 - 22x + 1 - 15 = 0 \implies x^2 - 22x - 14 = 0
\]

2. Next, we want to complete the square for the expression \( x^2 - 22x \). We take half of the coefficient of \( x \), which is \(-22\), divide by 2 to get \(-11\), and then square it to add and subtract \( 121 \) (which is \( (-11)^2 \)):
\[
x^2 - 22x + 121 - 121 - 14 = 0
\]
This simplifies to:
\[
(x - 11)^2 - 135 = 0
\]

3. Now isolate the squared term:
\[
(x - 11)^2 = 135
\]

4. Take the square root of both sides:
\[
x - 11 = \pm \sqrt{135}
\]
Simplifying \( \sqrt{135} \):
\[
\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}
\]
Thus,
\[
x - 11 = \pm 3\sqrt{15}
\]

5. Finally, solve for \( x \):
\[
x = 11 \pm 3\sqrt{15}
\]

Thus, the solutions are:
\[
x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15}
\]

The correct response is:
\[
x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15}
\]