Question

Complete the square to solve the following quadratic equation: x2−22x+1=15
.(1 point)
Responses

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+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 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 first rewrite the equation in standard form:

1. First, subtract 15 from both sides:
\[
x^2 - 22x + 1 - 15 = 0 \\
x^2 - 22x - 14 = 0
\]

2. Next, move the constant term to the right side:
\[
x^2 - 22x = 14
\]

3. To complete the square, we 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:
\[
(-11)^2 = 121
\]
Thus, we add 121 to both sides:
\[
x^2 - 22x + 121 = 14 + 121 \\
x^2 - 22x + 121 = 135
\]

4. Now, the left side can be factored as:
\[
(x - 11)^2 = 135
\]

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

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

7. We can simplify \( \sqrt{135} \):
\[
\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}
\]
So we have:
\[
x = 11 \pm 3\sqrt{15}
\]

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

Thus, the correct response is:
**x equals 11 plus 3 Start Root 15 End Root comma 11 minus 3 Start Root 15 End Root**.