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 will complete the square for the quadratic expression \( x^2 - 22x \).
-
Take half of the coefficient of \( x \) (which is -22), and square it: \[ \left(-\frac{22}{2}\right)^2 = (-11)^2 = 121 \]
-
Add and subtract 121 inside the equation:
\[ x^2 - 22x + 121 - 121 - 14 = 0 \] \[ (x - 11)^2 - 135 = 0 \]
Now, we'll move -135 to the other side:
\[ (x - 11)^2 = 135 \]
Next, we'll take the square root of both sides:
\[ x - 11 = \pm \sqrt{135} \]
Now, solving for \( x \):
\[ x = 11 \pm \sqrt{135} \]
Next, we simplify \( \sqrt{135} \):
\[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \]
This gives us:
\[ x = 11 \pm 3\sqrt{15} \]
So the solutions are:
\[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]
Now we can match these with the provided options.
Looking through the choices:
- A: \( x = -11 + 3\sqrt{15}, -11 - 3\sqrt{15} \)
- B: \( x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15} \)
- C: \( x = -11 + \sqrt{14}, -11 - \sqrt{14} \)
- D: \( x = 11 + \sqrt{14}, 11 - \sqrt{14} \)
The correct answer is:
B: \( x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15} \).