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

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

2. Add 14 to both sides: \[ x^2 - 22x = 14 \]

3. 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 \((-11)^2 = 121\).

   \[ x^2 - 22x + 121 = 14 + 121 \] \[ x^2 - 22x + 121 = 135 \]

4. Rewrite the left side as a square: \[ (x - 11)^2 = 135 \]

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

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

7. Simplify \( \sqrt{135} \): Notice that: \[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \]

   So we have: \[ x = 11 \pm 3\sqrt{15} \]

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

Answer: \( x=11 + 3\sqrt{15}, , 11 - 3\sqrt{15} \) is the correct choice.