Question

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

x=11+14−−√, 11−14−−√
x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 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+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+315−−√, 11−315−−√

Answer 1

To complete the square for the quadratic equation \( x^2 - 22x + 1 = 15 \), we first rearrange the equation to set it equal to zero:

\[
x^2 - 22x + 1 - 15 = 0
\]
\[
x^2 - 22x - 14 = 0
\]

Next, we can complete the square. We take the coefficient of \( x \) (which is \(-22\)), divide it by 2 (getting \(-11\)), and square it (getting \(121\)). We then add and subtract \(121\) in the equation:

\[
x^2 - 22x + 121 - 121 - 14 = 0
\]
\[
(x - 11)^2 - 135 = 0
\]

Next, we move \(135\) to the other side:

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

Now, we take the square root of both sides:

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

Then we solve for \(x\):

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

Thus, the solutions are:

\[
x = 11 + \sqrt{135} \quad \text{and} \quad x = 11 - \sqrt{135}
\]

The value of \(\sqrt{135}\) can be simplified to \(3\sqrt{15}\), so the final solutions can be expressed as:

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

Therefore, the correct responses are:

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