Question

Complete the square to solve the following quadratic equation: x^2 - 22x + 1 = 15.

A: x = -11 + 3 to the square root of 15, -11 -3 to the square root of 15.

B: x = 11 + 3 to the square root of 15, 11 - 3 to the square root of 15

C: x = -11 + to the square root of 14, -11 - to the square root of 14.

D: x = 11 + to the square root of 14, 11 - to the square root of 14

Answer 1

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 \( x^2 - 22x \).

1. Take half of the coefficient of \( x \) (which is -22), and square it:
\[
\left(-\frac{22}{2}\right)^2 = (-11)^2 = 121
\]

2. 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: \( x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15} \)**.