To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, follow these steps:
- Move the constant to the other side of the equation:
\[ x^2 - 22x + 1 - 15 = 0 \] \[ x^2 - 22x - 14 = 0 \]
- Rearrange the equation for completion of the square:
\[ x^2 - 22x = 14 \]
- Find the coefficient of \( x \), divide by 2, and square it:
The coefficient of \( x \) is \(-22\). Half of \(-22\) is \(-11\), and squaring it gives:
\[ (-11)^2 = 121 \]
- Add and subtract this square inside the equation:
\[ x^2 - 22x + 121 - 121 = 14 \] \[ (x - 11)^2 - 121 = 14 \]
- Now, move 121 to the other side:
\[ (x - 11)^2 = 14 + 121 \] \[ (x - 11)^2 = 135 \]
- Take the square root of both sides:
\[ x - 11 = \pm \sqrt{135} \]
This can be simplified since \( \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \):
\[ x - 11 = \pm 3\sqrt{15} \]
- Solve for \( x \):
\[ x = 11 \pm 3\sqrt{15} \]
Thus, the solutions are:
\[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]