To complete the square for the quadratic equation \( x^2 - 22x + 1 = 15 \), we first rearrange the equation to set it equal to zero:
\[ x^2 - 22x + 1 - 15 = 0 \] \[ x^2 - 22x - 14 = 0 \]
Next, we can complete the square. We take the coefficient of \( x \) (which is \(-22\)), divide it by 2 (getting \(-11\)), and square it (getting \(121\)). We then add and subtract \(121\) in the equation:
\[ x^2 - 22x + 121 - 121 - 14 = 0 \] \[ (x - 11)^2 - 135 = 0 \]
Next, we move \(135\) to the other side:
\[ (x - 11)^2 = 135 \]
Now, we take the square root of both sides:
\[ x - 11 = \pm \sqrt{135} \]
Then we solve for \(x\):
\[ x = 11 \pm \sqrt{135} \]
Thus, the solutions are:
\[ x = 11 + \sqrt{135} \quad \text{and} \quad x = 11 - \sqrt{135} \]
The value of \(\sqrt{135}\) can be simplified to \(3\sqrt{15}\), so the final solutions can be expressed as:
\[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]
Therefore, the correct responses are:
\[ x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15} \]