If `(sqrt(3)+1)^(2)=x+sqrt(3)y`, then the value of `(x+y)` is

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the equation \((\sqrt{3}+1)^{2}=x+\sqrt{3}y\), we will follow these steps:

### Step 1: Expand \((\sqrt{3}+1)^{2}\)
Using the formula for the square of a binomial, \((a+b)^{2} = a^{2} + 2ab + b^{2}\), we can expand \((\sqrt{3}+1)^{2}\).

\[ (\sqrt{3}+1)^{2} = (\sqrt{3})^{2} + 2(\sqrt{3})(1) + (1)^{2} \]

### Step 2: Calculate each term
Now, we calculate each term in the expansion:

1. \((\sqrt{3})^{2} = 3\)
2. \(2(\sqrt{3})(1) = 2\sqrt{3}\)
3. \((1)^{2} = 1\)

Putting these together, we have:

\[ (\sqrt{3}+1)^{2} = 3 + 2\sqrt{3} + 1 \]

### Step 3: Combine like terms
Now, we combine the constant terms:

\[ 3 + 1 = 4 \]

Thus, we have:

\[ (\sqrt{3}+1)^{2} = 4 + 2\sqrt{3} \]

### Step 4: Set the equation equal to \(x + \sqrt{3}y\)
Now, we can set this equal to \(x + \sqrt{3}y\):

\[ 4 + 2\sqrt{3} = x + \sqrt{3}y \]

### Step 5: Compare coefficients
From the equation \(4 + 2\sqrt{3} = x + \sqrt{3}y\), we can compare the coefficients of the constant terms and the coefficients of \(\sqrt{3}\):

1. The constant term gives us:
\[ x = 4 \]

2. The coefficient of \(\sqrt{3}\) gives us:
\[ y = 2 \]

### Step 6: Calculate \(x + y\)
Now, we can find \(x + y\):

\[ x + y = 4 + 2 = 6 \]

Thus, the final answer is:

\[ \boxed{6} \]