Question

Complete the square to solve the following quadratic equation: x^2−22x+1=15.(1 point)
Responses

x=−11+14−−√, −11−14−−√
x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root

x=11+315−−√, 11−315−−√
x equals 11 plus 3 Start Root 15 End Root comma 11 minus 3 Start Root 15 End Root

x=−11+315−−√, −11−315−−√
x equals negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 End Root

x=11+14−−√, 11−14−−√

Answer 1

To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, we can follow these steps:

1. **Rewrite the equation**:
\[
x^2 - 22x + 1 - 15 = 0
\]
This simplifies to:
\[
x^2 - 22x - 14 = 0
\]

2. **Move the constant term to the other side**:
\[
x^2 - 22x = 14
\]

3. **Complete the square**:
To complete the square for the left side, take half of the coefficient of \( x \) (which is -22), square it, and add it to both sides. Half of -22 is -11, and squaring it gives \( 121 \).
\[
x^2 - 22x + 121 = 14 + 121
\]
This simplifies to:
\[
(x - 11)^2 = 135
\]

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

5. **Solve for \( x \)**:
\[
x = 11 \pm \sqrt{135}
\]

6. **Factor \( \sqrt{135} \)**:
Since \( 135 = 9 \times 15 \), we can write:
\[
\sqrt{135} = 3\sqrt{15}
\]

7. **Final solutions**:
Therefore, the solutions can be written as:
\[
x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15}
\]

Thus, the correct response is:
**x=11+3√15, 11−3√15**.