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

Question

.(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+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−−√

Answer 1

To solve the equation \( x^2 - 22x + 1 = 15 \) by completing the square, we will follow these steps:

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

2. Complete the square on the left side. We need to take half of the coefficient of \( x \), which is \( -22 \), halve it to get \( -11 \), and then square it to get \( 121 \): \[ x^2 - 22x + 121 - 121 - 14 = 0 \] Simplifying: \[ (x - 11)^2 - 135 = 0 \]

3. Isolate the square: \[ (x - 11)^2 = 135 \]

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

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

Finally, we break down \( \sqrt{135} \): \[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \] So, we have: \[ x = 11 \pm 3\sqrt{15} \]

Thus, the solutions are: \[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]

The correct response is: x equals 11 plus 3√15, 11 minus 3√15.