Final answer:
To ensure both rectangles have the same area, we set their areas equal to each other and solve for x, which in this case yields x = 4 as the solution.
Step-by-step explanation:
To find the value of x so that two rectangles have the same area, we need to set the areas of both rectangles equal to each other. The area of a rectangle is found by multiplying its length by its width.
For the first rectangle with dimensions 2x by 2x, the area is (2x) × (2x), which simplifies to 4x². For the second rectangle with dimensions (x-2) cm by 32 cm, the area is (x-2) × 32 which simplifies to 32x - 64 cm².
Setting these two areas equal, we get:
4x² = 32x - 64
Now, we will solve this quadratic equation for x.
4x² - 32x + 64 = 0
Dividing the entire equation by 4 reduces it to:
x² - 8x + 16 = 0
Factoring the quadratic equation, we find:
(x - 4)(x - 4) = 0
So, x = 4 is the solution. This means that when x is 4, both rectangles will have the same area.