Question

Complete 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−−√
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 first rearrange the equation:

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

This simplifies to:

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

Next, we isolate the constant term on the other side:

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

Now, we complete the square on the left side. The coefficient of \( x \) is \(-22\). To complete the square, we take half of \(-22\), which is \(-11\), and then square it:

\[
(-11)^2 = 121
\]

Now, we add and subtract \( 121 \) on the left side:

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

This gives us:

\[
(x - 11)^2 - 121 = 14
\]

Adding \( 121 \) to both sides:

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

Now, we take the square root of both sides:

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

So,

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

The simplified form of \( \sqrt{135} \) can be further broken down:

\[
\sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15}
\]

Thus, we can express \( x \) as:

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

The final solutions are:

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

Hence, the correct response is:

**x=11+3√15, 11−3√15**.