Complete the square to solve the following quadratic equation: x2−22x+1=15.(1 point)

Question

Responses

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 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−−√

Answer 1

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

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

2. Now we will complete the square for the expression \( x^2 - 22x \). To do this, take half of the coefficient of \( x \), which is \(-22\), so half is \(-11\), and then square it: \[ \left(-11\right)^2 = 121 \]

3. Add and subtract this square inside the equation: \[ x^2 - 22x + 121 - 121 - 14 = 0 \] This can be rewritten as: \[ (x - 11)^2 - 135 = 0 \]

4. Rearranging gives: \[ (x - 11)^2 = 135 \]

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

6. Solve for \( x \) by adding 11 to both sides: \[ x = 11 \pm \sqrt{135} \]

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

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

Thus, the correct response is: x equals 11 plus 3√15, comma 11 minus 3√15.