If the equation `3x^(2)+6xy+my^(2)=0` represents a pair of coincident straight lines, then `(3m)/2` is equal to

Answer

To solve the problem, we need to determine the value of \(\frac{3m}{2}\) given that the equation \(3x^2 + 6xy + my^2 = 0\) represents a pair of coincident straight lines.

### Step-by-Step Solution:

1. **Understanding the Condition for Coincident Lines**:
The equation of a pair of coincident straight lines can be represented in the form:
\[ (y - m_1 x)(y - m_2 x) = 0 \]
where \(m_1\) and \(m_2\) are the slopes of the lines. For coincident lines, \(m_1 = m_2 = m'\).

2. **Rewriting the Equation**:
The given equation can be rewritten as:
\[ my^2 + 6xy + 3x^2 = 0 \]
This can be compared to the general form of a quadratic equation in \(y\):
\[ Ay^2 + Byx + Cx^2 = 0 \]
where \(A = m\), \(B = 6\), and \(C = 3\).

3. **Using the Condition for Coincident Lines**:
The condition for the equation to represent coincident lines is given by:
\[ B^2 - 4AC = 0 \]
Substituting the values of \(A\), \(B\), and \(C\):
\[ 6^2 - 4 \cdot m \cdot 3 = 0 \]
This simplifies to:
\[ 36 - 12m = 0 \]

4. **Solving for \(m\)**:
Rearranging the equation gives:
\[ 12m = 36 \]
Dividing both sides by 12:
\[ m = 3 \]

5. **Calculating \(\frac{3m}{2}\)**:
Now we can find \(\frac{3m}{2}\):
\[ \frac{3m}{2} = \frac{3 \cdot 3}{2} = \frac{9}{2} = 4.5 \]

### Final Answer:
Thus, the value of \(\frac{3m}{2}\) is:
\[ \boxed{4.5} \]