To find the minimum value for the equation [tex]\( 2x^2 - 4x - 2 = 0 \)[/tex], we need to complete the square.
First, let's start by dividing the entire equation by 2 to simplify the process:
[tex]\[ x^2 - 2x - 1 = 0 \][/tex]
Next, we'll complete the square for the expression [tex]\( x^2 - 2x \)[/tex]. To do this, we take half of the coefficient of [tex]\( x \)[/tex], which is [tex]\(-2\)[/tex], then we square it:
[tex]\[ \left(\frac{-2}{2}\right)^2 = 1 \][/tex]
Now, we add and subtract this square within the equation:
[tex]\[ x^2 - 2x + 1 - 1 - 1 = 0 \][/tex]
[tex]\[ (x - 1)^2 - 2 = 0 \][/tex]
Rearranging, we get:
[tex]\[ (x - 1)^2 - 2 = 0 \][/tex]
Next, let's consider the options provided:
1. [tex]\( 2(x-1)^2 = 4 \)[/tex]
2. [tex]\( 2(x-1)^2 = -4 \)[/tex]
3. [tex]\( 2(x-2)^2 = 4 \)[/tex]
4. [tex]\( 2(x-2)^2 = -4 \)[/tex]
We need to match our derived equation to one of the given forms. From our work, we see:
First, consider:
[tex]\[ 2(x - 1)^2 = 4 \][/tex]
[tex]\[ (x - 1)^2 = 2 \][/tex]
This does not match our equation.
Second, consider:
[tex]\[ 2(x - 1)^2 = -4 \][/tex]
[tex]\[ (x - 1)^2 = -2 \][/tex]
This does not match our equation.
Third, consider:
[tex]\[ 2(x - 2)^2 = 4 \][/tex]
[tex]\[ (x - 2)^2 = 2 \][/tex]
This does not match our equation.
Fourth, consider:
[tex]\[ 2(x - 2)^2 = -4 \][/tex]
[tex]\[ (x - 2)^2 = -2 \][/tex]
This does not match our equation.
It turns out upon reflection that we should simplify and correctly identify the transformation.
The correct transformation from our working should be:
[tex]\[ 2(x - 1)^2 = -4 \][/tex]
Therefore, out of the given options, the correct equation that reveals the minimum value is:
[tex]\[ \boxed{2(x-1)^2 = -4} \][/tex]
