Question:
If (a + b√3)^2 = 52 + 30√3, where a and b are natural numbers, find a + b.

Solution:

Expand the left side:
(a + b√3)^2 = a^2 + 2ab√3 + 3b^2

Equate real and irrational parts with the right side 52 + 30√3:

Real part:    a^2 + 3b^2 = 52
Irrational:   2ab = 30  ⇒  ab = 15

Find natural-number factor pairs of 15:
(ab = 15) ⇒ (a,b) ∈ {(1,15), (3,5), (5,3), (15,1)}

Check each pair against a^2 + 3b^2 = 52:

(1,15): 1^2 + 3·15^2 = 1 + 675 = 676 ≠ 52  
(3,5):  3^2 + 3·5^2  = 9 + 75  = 84  ≠ 52  
(5,3):  5^2 + 3·3^2  = 25 + 27 = 52  ✓  
(15,1): 15^2 + 3·1^2 = 225 + 3 = 228 ≠ 52

Only (a,b) = (5,3) satisfies both equations.

Therefore:
a + b = 5 + 3 = 8

Answer: 8