Complete the square to solve the following quadratic equation: x2−22x+1=15 .(1 point)

To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, follow these steps:

1. **Rearrange the equation**:
\[
x^2 - 22x + 1 - 15 = 0
\]
This simplifies to:
\[
x^2 - 22x - 14 = 0
\]

2. **Move the constant to the other side**:
\[
x^2 - 22x = 14
\]

3. **Complete the square**:
- Take half of the coefficient of \( x \) (which is -22), square it, and add it to both sides. Half of -22 is -11, and squaring it gives \( 121 \).
- So, add \( 121 \) to both sides:
\[
x^2 - 22x + 121 = 14 + 121
\]
This simplifies to:
\[
(x - 11)^2 = 135
\]

4. **Take the square root of both sides**:
\[
x - 11 = \pm \sqrt{135}
\]

5. **Solve for \( x \)**:
\[
x = 11 \pm \sqrt{135}
\]

6. **Simplify \( \sqrt{135} \)**:
\[
\sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15}
\]
Thus, we can write:
\[
x = 11 \pm 3\sqrt{15}
\]

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

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