Given information:
- The perimeter of a rectangle and square are equal.
- The area of the rectangle is 225 sq.m less than the square.

To find:
- The length and breadth of the rectangle.

Solution:

Let's assume the length and breadth of the rectangle as 'l' and 'b' respectively. Since the perimeter of the rectangle and square are equal, we can set up the equation:

2(l + b) = 4s

Here, 's' represents the side length of the square. Simplifying the equation, we get:

l + b = 2s

We are also given that the area of the rectangle is 225 sq.m less than the square. Mathematically, we can express this as:

lb = s^2 - 225

Substituting the value of 's' from the first equation:

lb = (l + b)^2 - 225

Expanding the square of the sum:

lb = l^2 + 2lb + b^2 - 225

Rearranging the terms:

l^2 + b^2 - 2lb = 225

Using the equation l + b = 2s:

l^2 + b^2 - 2lb = 225
l^2 + b^2 - 2l(2s - l) = 225
l^2 + b^2 - 4ls + 2l^2 = 225
3l^2 - 4ls + b^2 = 225

Since the perimeter of the rectangle and square are equal, the length of the square is 2l + 2b. Therefore, s = l + b.

Substituting the value of 's' in the above equation:

3l^2 - 4l(l + b) + b^2 = 225
3l^2 - 4l^2 - 4lb + b^2 = 225
-b^2 - 4lb + 225 = 0

Factoring the quadratic equation:

(b - 9)(b + 25) = 0

From this, we can deduce two possible values for 'b': b = 9 or b = -25. Since the breadth cannot be negative, we consider b = 9.

Substituting the value of 'b' in the equation l + b = 2s:

l + 9 = 2s
l = 2s - 9

Substituting the value of 'l' in the equation lb = s^2 - 225:

(2s - 9)(9) = s^2 - 225
18s - 81 = s^2 - 225
s^2 - 18s + 144 = 0
(s - 12)(s - 6) = 0

From this, we can deduce two possible values for 's': s = 12 or s = 6.

Calculating the length and breadth:

Case 1: When s = 12
l = 2(12) - 9 = 15
b = 9

Case 2: When s