If $(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$, where $a$ and $b$ are natural numbers, then $a + b$ equals

Given: $(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$, where $a$ and $b$ are natural numbers

Find: $a + b$

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Expand the left side using $(x + y)^2 = x^2 + 2xy + y^2$:

$(a + b\sqrt{3})^2 = a^2 + 2ab\sqrt{3} + 3b^2$

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Match both sides of the equation:

$a^2 + 3b^2 + 2ab\sqrt{3} = 52 + 30\sqrt{3}$

Since we have rational and irrational parts on both sides, we can equate the coefficients separately.

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Set up two equations:

Rational parts: $a^2 + 3b^2 = 52$ ... (1)

Irrational parts: $2ab = 30$ ... (2)

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From equation (2): $ab = 15$

Since $a$ and $b$ are natural numbers, $a$ must be a factor of $15$.

Factors of $15$: $1, 3, 5, 15$

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Test each possibility in equation (1):

If $a = 1, b = 15$: $1^2 + 3(15^2) = 1 + 675 = 676 \neq 52$

If $a = 3, b = 5$: $3^2 + 3(5^2) = 9 + 75 = 84 \neq 52$

If $a = 5, b = 3$: $5^2 + 3(3^2) = 25 + 27 = 52$ ✓

If $a = 15, b = 1$: $15^2 + 3(1^2) = 225 + 3 = 228 \neq 52$

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Therefore $a = 5$ and $b = 3$

$a + b = 5 + 3 = 8$