If `(3+2sqrt(5))^(2)=29+ksqrt(5)`, then what is the value of k?

Options

A. 12
B. 6
C. 29
D. 39

Answer

To solve the equation \((3 + 2\sqrt{5})^2 = 29 + k\sqrt{5}\), we will expand the left side and then compare the coefficients to find the value of \(k\).

### Step 1: Expand the left side
We start by expanding \((3 + 2\sqrt{5})^2\) using the formula for the square of a binomial, which states that \((a + b)^2 = a^2 + 2ab + b^2\).

Here, \(a = 3\) and \(b = 2\sqrt{5}\).

\[ (3 + 2\sqrt{5})^2 = 3^2 + 2 \cdot 3 \cdot (2\sqrt{5}) + (2\sqrt{5})^2 \]

### Step 2: Calculate each term
Now we calculate each term separately:

1. \(3^2 = 9\)
2. \(2 \cdot 3 \cdot (2\sqrt{5}) = 12\sqrt{5}\)
3. \((2\sqrt{5})^2 = 4 \cdot 5 = 20\)

### Step 3: Combine the results
Now, we combine these results:

\[ (3 + 2\sqrt{5})^2 = 9 + 20 + 12\sqrt{5} \]

This simplifies to:

\[ 29 + 12\sqrt{5} \]

### Step 4: Set the equation
Now we set the expanded form equal to the right side of the original equation:

\[ 29 + 12\sqrt{5} = 29 + k\sqrt{5} \]

### Step 5: Compare coefficients
To find \(k\), we compare the coefficients of \(\sqrt{5}\) on both sides of the equation:

From the left side, the coefficient of \(\sqrt{5}\) is \(12\), and from the right side, it is \(k\).

Thus, we have:

\[ k = 12 \]

### Final Answer
The value of \(k\) is \(12\).

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