Let x be the length of a side of the smaller square. Then the length of a side of the larger square is x + 25 (since the perimeter of the larger square exceeds that of the smaller by 100 cm, and each side is increased by 25 cm).
The area of the smaller square is x^2, and the area of the larger square is (x + 25)^2. We are told that the area of the larger square exceeds three times that of the smaller square by 325 cmsq, so we can write:
(x + 25)^2 = 3x^2 + 325
Expanding the left side, we get:
x^2 + 50x + 625 = 3x^2 + 325
Subtracting 2x^2 and 50x from both sides, we get:
275 = x^2 - 50x + 625
Simplifying further, we get:
0 = x^2 - 50x + 350
We can solve for x using the quadratic formula:
x = [50 ± sqrt(50^2 - 4(1)(350))] / (2(1))
x = [50 ± sqrt(2500 - 1400)] / 2
x = [50 ± sqrt(1100)] / 2
We can simplify this to:
x = 25 ± 5sqrt(11)
Since x represents the length of a side of a square, we can discard the negative solution. Therefore, the length of a side of the smaller square is:
x = 25 + 5sqrt(11) ≈ 48.8 cm
And the length of a side of the larger square is:
x + 25 ≈ 73.8 cm