Question

Complete the square to solve the following quadratic equation: x^2-22x+1=15

A. x=11+3√15,11-3√15

B. x=-11+3√15,-11-3√15

C. x=-11+√14,-11-√14

D. x=11+√14,11-√14

Answer 1

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

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

This simplifies to:

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

Next, to complete the square, we take the coefficient of \( x \), which is \( -22 \), divide it by \( 2 \) and square it:

\[
\left(-\frac{22}{2}\right)^2 = (-11)^2 = 121
\]

Now we can rewrite the equation by adding and subtracting \( 121 \):

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

This simplifies to:

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

Next, we set it equal to zero:

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

Taking the square root of both sides, we have:

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

We can simplify \( \sqrt{135} \):

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

Therefore, we rewrite the equation:

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

Adding \( 11 \) to both sides gives:

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

Thus, the solutions are:

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

The correct answer is:

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