To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, we first rearrange the equation:
\[ x^2 - 22x + 1 - 15 = 0 \]
This simplifies to:
\[ x^2 - 22x - 14 = 0 \]
Next, we isolate the constant term on the other side:
\[ x^2 - 22x = 14 \]
Now, we complete the square on the left side. The coefficient of \( x \) is \(-22\). To complete the square, we take half of \(-22\), which is \(-11\), and then square it:
\[ (-11)^2 = 121 \]
Now, we add and subtract \( 121 \) on the left side:
\[ x^2 - 22x + 121 - 121 = 14 \]
This gives us:
\[ (x - 11)^2 - 121 = 14 \]
Adding \( 121 \) to both sides:
\[ (x - 11)^2 = 135 \]
Now, we take the square root of both sides:
\[ x - 11 = \pm \sqrt{135} \]
So,
\[ x = 11 \pm \sqrt{135} \]
The simplified form of \( \sqrt{135} \) can be further broken down:
\[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \]
Thus, we can express \( x \) as:
\[ x = 11 \pm 3\sqrt{15} \]
The final solutions are:
\[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]
Hence, the correct response is:
x=11+3√15, 11−3√15.