Expand : (i) `(2x + (1)/( 2x))^2`

Answer

To expand the expression \((2x + \frac{1}{2x})^2\), we will use the algebraic identity for the square of a binomial, which states that:

\[ (a + b)^2 = a^2 + 2ab + b^2 \]

### Step-by-Step Solution:

1. **Identify \(a\) and \(b\)**:
- Let \(a = 2x\)
- Let \(b = \frac{1}{2x}\)

2. **Apply the identity**:
Using the identity \((a + b)^2 = a^2 + 2ab + b^2\), we can expand the expression:
\[ (2x + \frac{1}{2x})^2 = (2x)^2 + 2(2x)(\frac{1}{2x}) + \left(\frac{1}{2x}\right)^2 \]

3. **Calculate each term**:
- Calculate \(a^2\):
\[ (2x)^2 = 4x^2 \]
- Calculate \(2ab\):
\[ 2(2x)(\frac{1}{2x}) = 2 \cdot 2 \cdot \frac{1}{2} = 2 \]
- Calculate \(b^2\):
\[ \left(\frac{1}{2x}\right)^2 = \frac{1}{4x^2} \]

4. **Combine the terms**:
Now, substitute back into the expanded form:
\[ (2x + \frac{1}{2x})^2 = 4x^2 + 2 + \frac{1}{4x^2} \]

5. **Final expression**:
The final expanded expression is:
\[ 4x^2 + 2 + \frac{1}{4x^2} \]

### Final Answer:
\[ (2x + \frac{1}{2x})^2 = 4x^2 + 2 + \frac{1}{4x^2} \]