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

Question

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+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 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 equation \( x^2 - 22x + 1 = 15 \) by completing the square, we first rearrange the equation:

1. Move 15 to the left side: \[ x^2 - 22x + 1 - 15 = 0 \implies x^2 - 22x - 14 = 0 \]

2. Next, we complete the square. Start with the quadratic part, \( x^2 - 22x \).

   • Take \(-22\), divide by 2 to get \(-11\), and then square it to find \(121\).
   • Add and subtract \(121\) in the equation: \[ x^2 - 22x + 121 - 121 - 14 = 0 \implies (x - 11)^2 - 135 = 0 \]

3. Now, isolate the squared term: \[ (x - 11)^2 = 135 \]

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

5. Finally, solve for \(x\): \[ x = 11 \pm \sqrt{135} \]

6. We can simplify \(\sqrt{135}\): \[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \] Thus, the solutions become: \[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]

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