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 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+14−−√, 11−14−−√

Answer 1

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

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

This simplifies to:

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

Now, we'll complete the square for the expression \( x^2 - 22x \).

1. 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 -11 gives \( 121 \).

2. Rewrite the equation by adding \( 121 \) to both sides:

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

This simplifies to:

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

3. Now take the square root of both sides:

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

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

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

Now let's simplify \( \sqrt{135} \):

\[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \]

So, we have:

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

Thus, the solutions are:

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

The correct response from the options provided is:

x = 11 + 3√15, 11 - 3√15.