A square and a rectangle have the same perimeter p and their areas differ by q units. What is the square of the difference between the length and breadth of the rectangle?

Let side of square = s = p/4, so square area = s². Let rectangle have length l and breadth b, with l + b = p/2 = 2s. Area of rectangle = lb. Difference = s² − lb = q. Now (l − b)² = (l + b)² − 4lb = (2s)² − 4lb = 4s² − 4lb = 4(s² − lb) = 4q.