To solve this problem, let's first understand the conditions and variables involved:
1. Grid Size and Total Area: The given grid has dimensions 10 by 10, which means it consists of a total of [tex]\(10 \times 10 = 100\)[/tex] unit squares.
2. New Squares Side Lengths: We need to create two new squares using these 100 unit squares such that the side length of one square is 2 units greater than the side length of the other.
Let's define:
- [tex]\( x \)[/tex] as the side length of the greater square.
- [tex]\( x - 2 \)[/tex] as the side length of the smaller square (since it is 2 units less than [tex]\( x \)[/tex]).
Next, we need to express the total area of these squares using their side lengths:
- The area of the greater square will be [tex]\( x^2 \)[/tex].
- The area of the smaller square will be [tex]\((x - 2)^2\)[/tex].
Since the total area must add up to 100 unit squares, we can set up the following equation:
[tex]\[ x^2 + (x - 2)^2 = 100 \][/tex]
This equation represents the relationship between the side lengths of the two squares and the total area. Let's verify the options provided:
1. [tex]\( x^2 + (x - 2)^2 = 10 \)[/tex]
2. [tex]\( x^2 + 2 x^2 = 10 \)[/tex]
3. [tex]\( x^2 + (x - 2)^2 = 100 \)[/tex]
4. [tex]\( x^2 + 2 x^2 = 100 \)[/tex]
From our setup, we identified that the correct equation must be [tex]\( x^2 + (x - 2)^2 = 100 \)[/tex], which matches the third option.
So, the correct equation that represents [tex]\( x \)[/tex], the side length of the greater square, is:
[tex]\[ x^2 + (x - 2)^2 = 100 \][/tex]
