Question

C
omplete the square to solve the following quadratic equation: x2−22x+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−−√

Answer 1

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

1. Start by moving the constant term to the right side of the equation:

\[
x^2 - 22x + 1 - 15 = 0
\]

Simplifying this gives:

\[
x^2 - 22x - 14 = 0
\]

2. Next, complete the square. Take half of the coefficient of \(x\) (which is \(-22\)), square it, and add it inside the equation:

Half of \(-22\) is \(-11\), and squaring it gives \(121\). So we adjust the equation as follows:

\[
x^2 - 22x + 121 - 121 - 14 = 0
\]

This simplifies to:

\[
(x - 11)^2 - 135 = 0
\]

3. Rearranging gives us:

\[
(x - 11)^2 = 135
\]

4. Now, take the square root of both sides:

\[
x - 11 = \pm \sqrt{135}
\]

5. Finally, solve for \(x\):

\[
x = 11 \pm \sqrt{135}
\]

Since \(\sqrt{135} = 3\sqrt{15}\), we can rewrite it as:

\[
x = 11 \pm 3\sqrt{15}
\]

So, the correct responses from the options given would correspond to:

\[
x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15}
\]

Thus, the correct answer is:

**x = 11 + 3√15 , 11 - 3√15**.