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 11 plus 3 Start Root 15 End Root comma 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 negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 End Root

Answer 1

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

1. Start by moving all terms to one side of the equation:
\[
x^2 - 22x + 1 - 15 = 0
\]
This simplifies to:
\[
x^2 - 22x - 14 = 0
\]

2. Next, isolate the \( x^2 \) and \( x \) terms:
\[
x^2 - 22x = 14
\]

3. Complete the square on the left side. Take the coefficient of \( x \), which is \(-22\), divide it by 2 to get \(-11\), and then square it:
\[
(-11)^2 = 121
\]

4. Add \( 121 \) to both sides of the equation:
\[
x^2 - 22x + 121 = 14 + 121
\]
Simplifying the right side gives:
\[
x^2 - 22x + 121 = 135
\]

5. The left side can now be factored as a perfect square:
\[
(x - 11)^2 = 135
\]

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

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

Thus, the solutions are:
\[
x = 11 + \sqrt{135}, \quad x = 11 - \sqrt{135}
\]

The numerical simplification of \( \sqrt{135} \) is \( 3\sqrt{15} \), so we can also express the solutions as:
\[
x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15}
\]

Based on the options given, the correct response is:
**x equals 11 plus 3√15, 11 minus 3√15**.