To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, we first move \( 15 \) to the left side of the equation:
\[ x^2 - 22x + 1 - 15 = 0 \]
This simplifies to:
\[ x^2 - 22x - 14 = 0 \]
Next, to complete the square, we take the coefficient of \( x \), which is \( -22 \), divide it by \( 2 \) and square it:
\[ \left(-\frac{22}{2}\right)^2 = (-11)^2 = 121 \]
Now we can rewrite the equation by adding and subtracting \( 121 \):
\[ x^2 - 22x + 121 - 121 - 14 = 0 \]
This simplifies to:
\[ (x - 11)^2 - 135 = 0 \]
Next, we set it equal to zero:
\[ (x - 11)^2 = 135 \]
Taking the square root of both sides, we have:
\[ x - 11 = \pm \sqrt{135} \]
We can simplify \( \sqrt{135} \):
\[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \]
Therefore, we rewrite the equation:
\[ x - 11 = \pm 3\sqrt{15} \]
Adding \( 11 \) to both sides gives:
\[ x = 11 \pm 3\sqrt{15} \]
Thus, the solutions are:
\[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]
The correct answer is:
A. \( x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15} \)