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Complete the square to solve the following quadratic equation: x^2−22x+1=15 .(1 point)

Responses

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−−√
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 negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 End Root

x=11+315−−√, 11−315−−√

1 answer

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

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

  2. Next, we want to complete the square for the expression \( x^2 - 22x \). We take half of the coefficient of \( x \), which is \(-22\), divide by 2 to get \(-11\), and then square it to add and subtract \( 121 \) (which is \( (-11)^2 \)): \[ x^2 - 22x + 121 - 121 - 14 = 0 \] This simplifies to: \[ (x - 11)^2 - 135 = 0 \]

  3. Now isolate the squared term: \[ (x - 11)^2 = 135 \]

  4. Take the square root of both sides: \[ x - 11 = \pm \sqrt{135} \] Simplifying \( \sqrt{135} \): \[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \] Thus, \[ x - 11 = \pm 3\sqrt{15} \]

  5. Finally, solve for \( x \): \[ x = 11 \pm 3\sqrt{15} \]

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

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