To solve the quadratic equation \(x^2 - 22x + 1 = 15\) by completing the square, we first rewrite the equation in standard form:
\[ x^2 - 22x + 1 - 15 = 0 \]
This simplifies to:
\[ x^2 - 22x - 14 = 0 \]
Next, we move the constant to the right side:
\[ x^2 - 22x = 14 \]
Now, to complete the square, we take the coefficient of \(x\), which is \(-22\), divide by 2, and square it:
\[ \left(-\frac{22}{2}\right)^2 = (-11)^2 = 121 \]
We add this square on both sides:
\[ x^2 - 22x + 121 = 14 + 121 \]
This simplifies to:
\[ (x - 11)^2 = 135 \]
Next, we take the square root of both sides:
\[ x - 11 = \pm \sqrt{135} \]
Thus, we solve for \(x\):
\[ x = 11 \pm \sqrt{135} \]
We can simplify \(\sqrt{135}\):
\[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \]
So the solutions for \(x\) are:
\[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]
Hence, the correct response is:
x equals \(11 + 3\sqrt{15}, 11 - 3\sqrt{15}\).