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

Question

.(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 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−−√

Answer 1

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

1. Rearrange the equation: \[ x^2 - 22x + 1 - 15 = 0 \] which simplifies to: \[ x^2 - 22x - 14 = 0 \]

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

3. Complete the square. To do this, take half of the coefficient of \( x \) (which is -22), square it, and then add it to both sides. Half of -22 is -11, and squaring it gives 121: \[ x^2 - 22x + 121 = 14 + 121 \]

4. This simplifies to: \[ (x - 11)^2 = 135 \]

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

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

Now, simplify \( \sqrt{135} \):

• \( 135 = 9 \times 15 = 3^2 \times 15 \), thus \( \sqrt{135} = 3\sqrt{15} \).

7. Therefore, we find: \[ x = 11 \pm 3\sqrt{15} \]

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

Among the options provided, the correct response is: x = 11 + 3√15, 11 - 3√15.