Sure, let’s complete the square for the given expression [tex]\(x^2 - 2x - 2\)[/tex] and find its equivalent form among the given options.
The method of completing the square involves transforming a quadratic expression into the form [tex]\(a(x-h)^2 + k\)[/tex].
Begin with the given expression:
[tex]\[ x^2 - 2x - 2 \][/tex]
Step 1: Isolate the constant term.
[tex]\[ x^2 - 2x \][/tex]
Step 2: To complete the square, we 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]
Step 3: Add and subtract this square inside the expression:
[tex]\[ x^2 - 2x + 1 - 1 - 2 \][/tex]
Step 4: Now, the first three terms form a perfect square:
[tex]\[ (x - 1)^2 - 1 - 2 \][/tex]
Step 5: Simplify the constant terms:
[tex]\[ (x - 1)^2 - 3 \][/tex]
Thus, the expression [tex]\(x^2 - 2x - 2\)[/tex] can be written as:
[tex]\[ (x - 1)^2 - 3 \][/tex]
Comparing this with the given options:
[tex]\[
\begin{align*}
1. & \quad (x-1)^2 + 3 \\
2. & \quad (x-1)^2 - 3 \\
3. & \quad (x-2)^2 - 3 \\
4. & \quad (x-2)^2 + 3 \\
\end{align*}
\][/tex]
The matching form is:
[tex]\[ (x-1)^2 - 3 \][/tex]
So the correct option is:
[tex]\[ (x - 1)^2 - 3 \][/tex]
Therefore, the completed square form of the expression [tex]\(x^2 - 2x - 2\)[/tex] is [tex]\((x - 1)^2 - 3\)[/tex], which corresponds to option 2.
