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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 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 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

1 answer

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

  1. Rearrange the equation: Move all terms to one side of the equation. \[ x^2 - 22x + 1 - 15 = 0 \implies x^2 - 22x - 14 = 0 \]

  2. Isolate the constant term: Move the -14 to the right side of the equation. \[ x^2 - 22x = 14 \]

  3. Complete the square: 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. Take the square root: \[ x - 11 = \pm \sqrt{135} \]

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

  6. Simplify \( \sqrt{135} \): We can simplify \( \sqrt{135} \) as follows: \[ \sqrt{135} = \sqrt{9 \cdot 15} = \sqrt{9} \cdot \sqrt{15} = 3\sqrt{15} \]

Thus, our solutions can be expressed as: \[ x = 11 \pm 3\sqrt{15} \]

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

So the correct choice is: x=11+3√15, 11−3√15.