Question

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

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+315−−√, 11−315−−√
x equals 11 plus 3 Start Root 15 End Root comma 11 minus 3 Start Root 15 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−−√
x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 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'll isolate the \(x\) terms:

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

Now we complete the square. The coefficient of \(x\) is \(-22\), so we take half of that value, square it, and add it to both sides. Half of \(-22\) is \(-11\), and squaring it gives \(121\).

Add \(121\) to both sides:

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

This gives us:

\[
x^2 - 22x + 121 = 135
\]

Now we can write the left side as a squared binomial:

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

Next, we take the square root of both sides:

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

Now, we solve for \(x\):

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

Finally, we simplify \(\sqrt{135}\):

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

So our solutions are:

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

Thus, the correct response is:

**x = 11 + 3√15, 11 - 3√15**.