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

Let the side of the square be s, so 4s = p, giving s = p/4. Let the rectangle have length l and breadth b, with 2(l + b) = p, so l + b = p/2 = 2s. Area of square = s², 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.