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Complete the square to solve the following quadratic equation: x2−22x+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 negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root

x=11+14−−√, 11−14−−√

1 answer

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

  1. First, rearrange the equation: \[ x^2 - 22x + 1 - 15 = 0 \] This simplifies to: \[ x^2 - 22x - 14 = 0 \]

  2. Next, isolate the quadratic and linear terms on one side: \[ x^2 - 22x = 14 \]

  3. 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: \[ x^2 - 22x + 121 = 14 + 121 \] Thus, we have: \[ (x - 11)^2 = 135 \]

  4. Now, take the square root of both sides: \[ x - 11 = \pm \sqrt{135} \]

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

Now, let's express \( \sqrt{135} \): \[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \]

So, the solutions can be written as: \[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]

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

This corresponds to the first option: x equals 11 plus 3 sqrt(15), 11 minus 3 sqrt(15).