### Step-by-Step Solution:
#### 1) Rewrite the equation by completing the square.
To complete the square for the quadratic equation [tex]\(x^2 - 2x - 6 = 0\)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[
x^2 - 2x - 6 = 0
\][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[
x^2 - 2x = 6
\][/tex]
3. To complete the square, take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-2\)[/tex], divide it by 2 and then square it:
[tex]\[
\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1
\][/tex]
4. Add this square to both sides of the equation:
[tex]\[
x^2 - 2x + 1 = 6 + 1
\][/tex]
5. Now the left side of the equation becomes a perfect square:
[tex]\[
(x - 1)^2 = 7
\][/tex]
So, the equation in the completed square form is:
[tex]\[
(x - 1)^2 = 7
\][/tex]
#### 2) What are the solutions to the equation?
To find the solutions, take the square root of both sides:
[tex]\[
x - 1 = \pm \sqrt{7}
\][/tex]
This gives us two solutions:
[tex]\[
x = 1 \pm \sqrt{7}
\][/tex]
#### Therefore, the correct answer is:
(A) [tex]\( x = 1 \pm \sqrt{7} \)[/tex]
