Which of the following is equivalent to `(a+b/2)^2` ?

Options

A. `a^2 + b^2/2`
B. `a^2 + b^2/4`
C. `a^2 + (ab)/2 + b^2/2`
D. `a^2 + ab + b^2/4`

Explanation

To find the expression equivalent to \((a + \frac{b}{2})^2\), we will expand it using the formula for the square of a binomial.

### Step-by-Step Solution:

1. **Identify the formula**: We will use the formula for the square of a sum, which is given by:
\[ (x + y)^2 = x^2 + 2xy + y^2 \]
Here, we will let \(x = a\) and \(y = \frac{b}{2}\).

2. **Substitute \(x\) and \(y\)**: Substitute \(x\) and \(y\) into the formula:
\[ (a + \frac{b}{2})^2 = a^2 + 2 \cdot a \cdot \frac{b}{2} + \left(\frac{b}{2}\right)^2 \]

3. **Calculate each term**:
- The first term is:
\[ a^2 \]
- The second term is:
\[ 2 \cdot a \cdot \frac{b}{2} = ab \]
- The third term is:
\[ \left(\frac{b}{2}\right)^2 = \frac{b^2}{4} \]

4. **Combine all terms**: Now, we combine all the calculated terms:
\[ a^2 + ab + \frac{b^2}{4} \]

5. **Final expression**: Therefore, the expression equivalent to \((a + \frac{b}{2})^2\) is:
\[ a^2 + ab + \frac{b^2}{4} \]

### Final Answer:
\[ a^2 + ab + \frac{b^2}{4} \]