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

A. 7
B. 10
C. 8
D. 9

Question

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

A. 7  
B. 10  
C. 8  
D. 9

Accepted Answer

The given equation is:  
$$
(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}
$$

Expanding the left-hand side:  
$$
a^2 + 2ab\sqrt{3} + 3b^2 = 52 + 30\sqrt{3}
$$

By comparing the rational and irrational parts:  
1. From the rational part:  
   $$
   a^2 + 3b^2 = 52
   $$
2. From the irrational part:  
   $$
   2ab = 30 \implies ab = 15
   $$

Now, solve the system of equations:
1. $$ a^2 + 3b^2 = 52 $$  
2. $$ ab = 15 $$

### Step 1: Solve for possible values of $$a$$ and $$b$$
Since $$a$$ and $$b$$ are natural numbers, consider factor pairs of $$ab = 15$$:  
$$
(a, b) = (1, 15), (15, 1), (3, 5), (5, 3)
$$

### Step 2: Check which pair satisfies $$a^2 + 3b^2 = 52$$
- For $$a = 5, b = 3$$:  
   $$
   a^2 + 3b^2 = 5^2 + 3(3^2) = 25 + 27 = 52
   $$
   This satisfies the equation.

- For other pairs, the equation does not hold.

### Step 3: Calculate $$a + b$$
The valid solution is $$a = 5, b = 3$$. Therefore:  
$$
a + b = 5 + 3 = 8
$$

### Final Answer:
**Option C: 8**

Answer

Answer

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