Question

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

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

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

Answer 1

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

1. **Rearrange the equation**: Move all terms to one side of the equation.
\[
x^2 - 22x + 1 - 15 = 0 \implies x^2 - 22x - 14 = 0
\]

2. **Isolate the constant term**: Move the -14 to the right side of the equation.
\[
x^2 - 22x = 14
\]

3. **Complete the square**: To complete the square, 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**:
\[
x - 11 = \pm \sqrt{135}
\]

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

6. **Simplify \( \sqrt{135} \)**:
We can simplify \( \sqrt{135} \) as follows:
\[
\sqrt{135} = \sqrt{9 \cdot 15} = \sqrt{9} \cdot \sqrt{15} = 3\sqrt{15}
\]

Thus, our solutions can be expressed as:
\[
x = 11 \pm 3\sqrt{15}
\]

Therefore, the correct response is:
\[
x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15}
\]

So the correct choice is:
**x=11+3√15, 11−3√15**.