To transform the quadratic expression [tex]\(x^2 - 2x - 2\)[/tex] into the form [tex]\(a(x - h)^2 + k\)[/tex], we shall complete the square. Here are the detailed steps:
1. Start with the given expression:
[tex]\[
x^2 - 2x - 2
\][/tex]
2. Complete the square for the quadratic and linear terms:
First, we focus on [tex]\(x^2 - 2x\)[/tex]. To complete the square, we need to add and subtract the square of half the coefficient of [tex]\(x\)[/tex]. The coefficient of [tex]\(x\)[/tex] is [tex]\(-2\)[/tex], so half of it is [tex]\(-1\)[/tex], and squaring it gives:
[tex]\[
\left( \frac{-2}{2} \right)^2 = (-1)^2 = 1
\][/tex]
Add and subtract this square within the expression:
[tex]\[
x^2 - 2x + 1 - 1 - 2
\][/tex]
3. Rewrite the expression as a completed square and a constant:
[tex]\[
(x - 1)^2 - 1 - 2
\][/tex]
Now combine the constants:
[tex]\[
(x - 1)^2 - 3
\][/tex]
4. Compare with the given options:
- Option 1: [tex]\( (x - 1)^2 + 3 \)[/tex]
- Option 2: [tex]\( (x - 1)^2 - 3 \)[/tex]
- Option 3: [tex]\( (x - 2)^2 - 3 \)[/tex]
- Option 4: [tex]\( (x - 2)^2 + 3 \)[/tex]
We see that the transformed expression [tex]\( (x - 1)^2 - 3 \)[/tex] matches Option 2.
Therefore, the correct transformation that completes the square for the expression [tex]\(x^2 - 2x - 2\)[/tex] is:
[tex]\[
(x - 1)^2 - 3
\][/tex]
So the correct choice is:
[tex]\[
\boxed{(x - 1)^2 - 3}
\][/tex] or simply Option 2.
