Patricia purchased x meters of fencing. She originally intended to use all of the fencing to enclose a square region, but later decided to use all of the fencing to enclose a rectangular region with length y meters greater than its width. In square meters, what is the positive difference between the area of the square region and the area of the rectangular region?

(y^2) / 4 square meters

Step-by-step explanation:

For a perimeter length of x, the side of a square will be x/4 and its area will be (x/4) ^2.

If one side of the square is shortened by y/2 and the adjacent side is lengthened by y/2, then the difference in side lengths will be y. The area of the resulting rectangle will be ...

(x/4 - y/2) (x/4 + y/2) = (x/4) ^2 - (y/2) ^2

That is, the difference in area between the square and the rectangle is ...

(x/4) ^2 - ((x/4) ^2 - (y/2) ^2) = (y/2) ^2 = y^2/4

The positive difference between the area of the square region and the area of the rectangular region is y^2/4 square meters.