Question

Complete the square to solve the following quadratic equation: x^2−22x+1=15
x= -11 + 3√15, -11 -3√15
x=11+ 3√15, 11- 3√15
x=11+√14, 11 - √14
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:

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

This simplifies to:

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

Now, we will complete the square. Start by rearranging the equation:

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

Next, find the term 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 \(121\):

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

This simplifies to:

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

Next, take the square root of both sides:

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

This further simplifies to:

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

Therefore, we can isolate \( x \):

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

The solutions to the original equation are:

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

Now, looking at the options you provided, the correct answer is:

**\(x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15}\)**