Question

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

Answer 1

To solve the quadratic equation \(x^2 - 22x + 1 = 15\) by completing the square, we'll start by rearranging the equation.

First, subtract 15 from both sides:

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

This simplifies to:

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

Next, we want to complete the square. We'll focus on the \(x^2 - 22x\) part. To complete the square, take half of the coefficient of \(x\) (which is -22), square it, and add that value to both sides.

1. Half of -22 is -11.
2. Squaring -11 gives \(121\).

Now add 121 to both sides:

\[
x^2 - 22x + 121 - 135 = 0
\]

This can be rewritten as:

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

Now, isolate the squared term:

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

Next, take the square root of both sides:

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

Now, solve for \(x\):

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

Since \(135\) can be simplified as \(9 \times 15\) and \(\sqrt{9} = 3\):

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

So the solutions can be expressed as:

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

Thus, the complete solutions are:

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

The correct response is:
\[
x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15}
\]