Ask a New Question
Menu

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+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 negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 End Root

1 answer

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

  1. Start by moving all terms to one side of the equation: \[ x^2 - 22x + 1 - 15 = 0 \] This simplifies to: \[ x^2 - 22x - 14 = 0 \]

  2. Next, isolate the \( x^2 \) and \( x \) terms: \[ x^2 - 22x = 14 \]

  3. Complete the square on the left side. Take the coefficient of \( x \), which is \(-22\), divide it by 2 to get \(-11\), and then square it: \[ (-11)^2 = 121 \]

  4. Add \( 121 \) to both sides of the equation: \[ x^2 - 22x + 121 = 14 + 121 \] Simplifying the right side gives: \[ x^2 - 22x + 121 = 135 \]

  5. The left side can now be factored as a perfect square: \[ (x - 11)^2 = 135 \]

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

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

Thus, the solutions are: \[ x = 11 + \sqrt{135}, \quad x = 11 - \sqrt{135} \]

The numerical simplification of \( \sqrt{135} \) is \( 3\sqrt{15} \), so we can also express the solutions as: \[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]

Based on the options given, the correct response is: x equals 11 plus 3√15, 11 minus 3√15.