Simplify:- (2+√5)^2 = ? + 4√5

Options

A. a. 9+4√5
B. b. 9
C. c. 2√5
D. d. 9+2√5

Answer

To simplify the expression \((2 + \sqrt{5})^2\) and find the value represented by the question mark in the equation \((2 + \sqrt{5})^2 = ? + 4\sqrt{5}\), we can follow these steps:

### Step 1: Apply the formula for squaring a binomial
We use the formula \((a + b)^2 = a^2 + 2ab + b^2\). Here, let \(a = 2\) and \(b = \sqrt{5}\).

### Step 2: Calculate \(a^2\)
Calculate \(2^2\):
\[ 2^2 = 4 \]

### Step 3: Calculate \(b^2\)
Calculate \((\sqrt{5})^2\):
\[ (\sqrt{5})^2 = 5 \]

### Step 4: Calculate \(2ab\)
Calculate \(2 \cdot 2 \cdot \sqrt{5}\):
\[ 2 \cdot 2 \cdot \sqrt{5} = 4\sqrt{5} \]

### Step 5: Combine the results
Now, combine all the parts:
\[ (2 + \sqrt{5})^2 = a^2 + b^2 + 2ab = 4 + 5 + 4\sqrt{5} \]
This simplifies to:
\[ (2 + \sqrt{5})^2 = 9 + 4\sqrt{5} \]

### Step 6: Set the equation
Now we have:
\[ 9 + 4\sqrt{5} = ? + 4\sqrt{5} \]

### Step 7: Isolate the question mark
To find the value of the question mark, we can subtract \(4\sqrt{5}\) from both sides:
\[ 9 = ? \]

### Final Answer
Thus, the value represented by the question mark is:
\[ ? = 9 \]

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