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

Question

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 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 can follow these steps:

1. Start by rearranging the equation: \[ x^2 - 22x + 1 - 15 = 0 \] This simplifies to: \[ x^2 - 22x - 14 = 0 \]

2. Move \(-14\) 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 it gives \(121\): \[ x^2 - 22x + 121 = 14 + 121 \] This simplifies to: \[ (x - 11)^2 = 135 \]

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

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

Since \(\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}\), the solutions can be expressed as: \[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]

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

So the answer is: x=11+315−−√, 11−315−−√.