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Complete the square to solve the following quadratic equation: x^2−22x+1=15 .(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+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 11 plus Start Root 14 End Root comma 11 minus Start Root 14 End Root

x=−11+14−−√, −11−14−−√

1 answer

To solve the equation \( x^2 - 22x + 1 = 15 \) by completing the square, we can start by simplifying and rearranging the equation:

  1. Subtract 15 from both sides: \[ x^2 - 22x + 1 - 15 = 0 \implies x^2 - 22x - 14 = 0 \]

  2. Now, we will complete the square. We take the coefficient of \( x \), which is \(-22\), divide it by 2, and then square it: \[ \left( \frac{-22}{2} \right)^2 = (-11)^2 = 121 \]

  3. Rewrite the quadratic by adding and subtracting this square inside the equation: \[ x^2 - 22x + 121 - 121 - 14 = 0 \implies (x - 11)^2 - 135 = 0 \]

  4. Move the constant to the other side: \[ (x - 11)^2 = 135 \]

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

  6. Add 11 to both sides: \[ x = 11 \pm \sqrt{135} \]

Since \( \sqrt{135} = 3\sqrt{15} \), we can write: \[ x = 11 \pm 3\sqrt{15} \]

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

Therefore, the correct response is: \( x = 11 + 3 \sqrt{15}, ; 11 - 3 \sqrt{15} \).