Question

Complete the square to solve the following quadratic equation: x^2 - 22x + 1 = 15.
A) x = 11 + 3√15, 11 - 3√15
B) x = -11 + 3√15, - 11 - 3√15
C) x = -11 + √14, - 11 - √14
D) x = 11 + √14, 11 - √14

Answer 1

To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, we start by rearranging the equation:

1. Subtract 15 from both sides to get:
\[
x^2 - 22x + 1 - 15 = 0
\]
This simplifies to:
\[
x^2 - 22x - 14 = 0
\]

2. Now we need to complete the square for the quadratic expression \( x^2 - 22x \).

To complete the square, we take half of the coefficient of \( x \) (which is -22), square it, and then add and subtract that value inside the equation. Half of -22 is -11, and squaring -11 gives 121.

3. Rewrite the expression by completing the square:
\[
x^2 - 22x = (x - 11)^2 - 121
\]
Consequently, our equation becomes:
\[
(x - 11)^2 - 121 - 14 = 0
\]
Which simplifies to:
\[
(x - 11)^2 - 135 = 0
\]

4. Add 135 to both sides to isolate the squared term:
\[
(x - 11)^2 = 135
\]

5. Take the square root of both sides:
\[
x - 11 = \pm \sqrt{135}
\]

6. Solving for \( x \):
\[
x = 11 \pm \sqrt{135}
\]

7. To simplify \(\sqrt{135}\):
\[
\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}
\]
Thus, we have:
\[
x = 11 \pm 3\sqrt{15}
\]

The solutions are:
\[
x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15}
\]

Thus, the final answer corresponds to option A:
\[
\text{A) } x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15}
\]