C

Question

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

x=11+315−−√, 11−315−−√

Answer 1

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

1. Start by moving the constant term to the right side of the equation:

   \[ x^2 - 22x + 1 - 15 = 0 \]

   Simplifying this gives:

   \[ x^2 - 22x - 14 = 0 \]

2. Next, complete the square. Take half of the coefficient of \(x\) (which is \(-22\)), square it, and add it inside the equation:

   Half of \(-22\) is \(-11\), and squaring it gives \(121\). So we adjust the equation as follows:

   \[ x^2 - 22x + 121 - 121 - 14 = 0 \]

   This simplifies to:

   \[ (x - 11)^2 - 135 = 0 \]

3. Rearranging gives us:

   \[ (x - 11)^2 = 135 \]

4. Now, take the square root of both sides:

   \[ x - 11 = \pm \sqrt{135} \]

5. Finally, solve for \(x\):

   \[ x = 11 \pm \sqrt{135} \]

Since \(\sqrt{135} = 3\sqrt{15}\), we can rewrite it as:

\[ x = 11 \pm 3\sqrt{15} \]

So, the correct responses from the options given would correspond to:

\[ x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15} \]

Thus, the correct answer is:

x = 11 + 3√15 , 11 - 3√15.