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A group of students is given a 10 by 10 grid to cut into individual unit squares. The challenge is to create two squares using all of the unit squares. Their teacher states that after the two new squares are formed, one should have a side length two units greater than the other.

Which equation represents [tex]\( x \)[/tex], the side length of the greater square?

A. [tex]\( x^2 + (x - 2)^2 = 10 \)[/tex]
B. [tex]\( x^2 + 2x^2 = 10 \)[/tex]
C. [tex]\( x^2 + (x - 2)^2 = 100 \)[/tex]
D. [tex]\( x^2 + 2x^2 = 100 \)[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to find the equation that represents the side length [tex]\( x \)[/tex] of the greater square created from a 10 by 10 grid.

First, calculate the total area of the 10 by 10 grid:
[tex]\[ \text{Total area} = 10 \times 10 = 100 \][/tex]

Next, let's denote [tex]\( x \)[/tex] as the side length of the greater square. According to the problem, the side length of the smaller square is 2 units less than that of the greater square, so the side length of the smaller square is [tex]\( x - 2 \)[/tex].

Now let's write the equations for the areas of the two squares:
- The area of the greater square is [tex]\( x^2 \)[/tex].
- The area of the smaller square is [tex]\( (x - 2)^2 \)[/tex].

Since the two squares must use all of the area from the 10 by 10 grid, their combined areas should equal the total area of the grid:
[tex]\[ x^2 + (x - 2)^2 = 100 \][/tex]

Thus, the equation that represents [tex]\( x \)[/tex], the side length of the greater square, is:
[tex]\[ x^2 + (x - 2)^2 = 100 \][/tex]

So, the correct answer is:
[tex]\[ x^2 + (x - 2)^2 = 100 \][/tex]

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