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

Question

.(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 11 plus 3 Start Root 15 End Root comma 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 follow these steps:

1. First, rearrange the equation: \[ x^2 - 22x + 1 - 15 = 0 \] which simplifies to: \[ x^2 - 22x - 14 = 0 \]

2. Move the constant term to the other side: \[ x^2 - 22x = 14 \]

3. To complete the square, take half of the coefficient of \( x \) (which is -22), square it, and add it to both sides. Half of -22 is -11, and squaring -11 gives 121: \[ x^2 - 22x + 121 = 14 + 121 \] So, we have: \[ (x - 11)^2 = 135 \]

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

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

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

The final answer, matching the options provided, would be: \[ x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15} \]

Therefore, the correct response is: x=11+3√15, 11−3√15