To solve the quadratic equation \(x^2 - 22x + 1 = 15\) by completing the square, we'll start by rearranging the equation.
First, subtract 15 from both sides:
\[ x^2 - 22x + 1 - 15 = 0 \]
This simplifies to:
\[ x^2 - 22x - 14 = 0 \]
Next, we want to complete the square. We'll focus on the \(x^2 - 22x\) part. To complete the square, take half of the coefficient of \(x\) (which is -22), square it, and add that value to both sides.
- Half of -22 is -11.
- Squaring -11 gives \(121\).
Now add 121 to both sides:
\[ x^2 - 22x + 121 - 135 = 0 \]
This can be rewritten as:
\[ (x - 11)^2 - 135 = 0 \]
Now, isolate the squared term:
\[ (x - 11)^2 = 135 \]
Next, take the square root of both sides:
\[ x - 11 = \pm \sqrt{135} \]
Now, solve for \(x\):
\[ x = 11 \pm \sqrt{135} \]
Since \(135\) can be simplified as \(9 \times 15\) and \(\sqrt{9} = 3\):
\[ \sqrt{135} = 3\sqrt{15} \]
So the solutions can be expressed as:
\[ x = 11 \pm 3\sqrt{15} \]
Thus, the complete solutions are:
\[ x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15} \]
The correct response is: \[ x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15} \]