Expand `(x-3y+(z)/(2))^(2)`.

Answer

To expand the expression \((x - 3y + \frac{z}{2})^2\), we will use the formula for the square of a trinomial, which is given by:

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

### Step-by-Step Solution:

1. **Identify \(a\), \(b\), and \(c\)**:
- Let \(a = x\)
- Let \(b = -3y\)
- Let \(c = \frac{z}{2}\)

2. **Calculate \(a^2\)**:
\[ a^2 = x^2 \]

3. **Calculate \(b^2\)**:
\[ b^2 = (-3y)^2 = 9y^2 \]

4. **Calculate \(c^2\)**:
\[ c^2 = \left(\frac{z}{2}\right)^2 = \frac{z^2}{4} \]

5. **Calculate \(2ab\)**:
\[ 2ab = 2 \cdot x \cdot (-3y) = -6xy \]

6. **Calculate \(2bc\)**:
\[ 2bc = 2 \cdot (-3y) \cdot \frac{z}{2} = -3yz \]

7. **Calculate \(2ca\)**:
\[ 2ca = 2 \cdot \frac{z}{2} \cdot x = zx \]

8. **Combine all the terms**:
Now, we can combine all the calculated terms:
\[ (x - 3y + \frac{z}{2})^2 = x^2 + 9y^2 + \frac{z^2}{4} - 6xy - 3yz + zx \]

### Final Expanded Form:
Thus, the expanded form of \((x - 3y + \frac{z}{2})^2\) is:
\[ x^2 + 9y^2 + \frac{z^2}{4} - 6xy - 3yz + zx \]