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

Question

.(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+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+14−−√, 11−14−−√

Answer 1

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

1. Subtract 15 from both sides: \[ x^2 - 22x + 1 - 15 = 0 \implies x^2 - 22x - 14 = 0 \]

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

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 \(121\): \[ x^2 - 22x + 121 = 14 + 121 \] \[ x^2 - 22x + 121 = 135 \]

4. Now, the left side can be factored as: \[ (x - 11)^2 = 135 \]

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

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

Now, simplify \(\sqrt{135}\): \[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \]

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

Based on the available responses, the correct answer is: x = 11 + 3√15, 11 - 3√15.