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+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 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 solve the quadratic equation \(x^2 - 22x + 1 = 15\) by completing the square, we first rewrite the equation in standard form:

1. Subtract 15 from both sides:
\[
x^2 - 22x + 1 - 15 = 0
\]
This simplifies to:
\[
x^2 - 22x - 14 = 0
\]

2. To complete the square, we need to manipulate the quadratic on the left side. Start with the terms involving \(x\):
\[
x^2 - 22x
\]

3. Take half of the coefficient of \(x\) (which is -22), square it, and add it inside the equation. Half of -22 is -11, and squaring it gives 121:

Rewrite the equation:
\[
x^2 - 22x + 121 - 121 - 14 = 0
\]
This can be rearranged to:
\[
(x - 11)^2 - 135 = 0
\]

4. Now isolate the perfect square:
\[
(x - 11)^2 = 135
\]

5. Take the square root of both sides:
\[
x - 11 = \pm \sqrt{135}
\]

6. Solve for \(x\):
\[
x = 11 \pm \sqrt{135}
\]

Thus, the solutions are:
\[
x = 11 + \sqrt{135}, \quad 11 - \sqrt{135}
\]

Now simplify \(\sqrt{135}\):
\[
\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}
\]

Finally, substitute this back:
\[
x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15}
\]

Thus, the correct answer choice is:
\[
x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15}
\]