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 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+14−−√, 11−14−−√

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 complete the square for the expression \( x^2 - 22x \). To do this, we take half of the coefficient of \( x \) (which is \(-22\)), square it, and add it inside the square:

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

We rewrite the equation by adding and subtracting \(121\):

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

This simplifies to:

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

Next, we set the equation to zero:

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

Taking the square root of both sides, we get:

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

Thus,

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

To simplify \( \sqrt{135} \):

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

Now, substituting back into our solutions, we have:

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

So the complete solutions are:

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

Thus, the correct response is:

**x equals 11 plus 3√15, 11 minus 3√15**