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

Question

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+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

Answer 1

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

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

2. Rearrange it for completing the square: \[ x^2 - 22x = 14 \]

3. 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 \(-11\).
   • Squaring \(-11\) gives \(121\).

   So, we add \(121\) to both sides: \[ x^2 - 22x + 121 = 14 + 121 \] \[ x^2 - 22x + 121 = 135 \]

4. Rewrite the left side as a square: \[ (x - 11)^2 = 135 \]

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

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

   Now we can simplify \( \sqrt{135} \): \[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \]

7. Substituting back gives: \[ x = 11 \pm 3\sqrt{15} \]

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

So the correct response from the options provided is: \[ x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15} \]

Thus, the answer is:

• \( x=11+3\sqrt{15}, ; 11−3\sqrt{15} \)