`(13+2sqrt(5))^(2)=?sqrt(5)+189`

Options

A. 26
B. 25
C. 52
D. 130

Explanation

To solve the equation \((13 + 2\sqrt{5})^2 = ?\sqrt{5} + 189\), we will follow these steps:

### Step 1: Expand the left side
We start by expanding the left side of the equation using the formula for the square of a binomial, \((a + b)^2 = a^2 + 2ab + b^2\).

Here, \(a = 13\) and \(b = 2\sqrt{5}\).

\[ (13 + 2\sqrt{5})^2 = 13^2 + 2 \cdot 13 \cdot (2\sqrt{5}) + (2\sqrt{5})^2 \]

Calculating each term:
- \(13^2 = 169\)
- \(2 \cdot 13 \cdot (2\sqrt{5}) = 52\sqrt{5}\)
- \((2\sqrt{5})^2 = 4 \cdot 5 = 20\)

Putting it all together:
\[ (13 + 2\sqrt{5})^2 = 169 + 52\sqrt{5} + 20 \]

### Step 2: Combine like terms
Now, we combine the constant terms:
\[ 169 + 20 = 189 \]
So, we have:
\[ (13 + 2\sqrt{5})^2 = 189 + 52\sqrt{5} \]

### Step 3: Set the equation
Now we can set the equation:
\[ 189 + 52\sqrt{5} = ?\sqrt{5} + 189 \]

### Step 4: Isolate the terms with \(\sqrt{5}\)
Subtract \(189\) from both sides:
\[ 52\sqrt{5} = ?\sqrt{5} \]

### Step 5: Solve for the question mark
Since both sides have \(\sqrt{5}\), we can equate the coefficients:
\[ ? = 52 \]

### Final Answer
Thus, the value of the question mark is:
\[ \boxed{52} \]

---