Determine the values of m for which the equations `3x^(2)+4mx+2=0 and 2x^(2)+3x-2=0` may have a common root.

Answer

To determine the values of \( m \) for which the equations \( 3x^2 + 4mx + 2 = 0 \) and \( 2x^2 + 3x - 2 = 0 \) have a common root, we can follow these steps:

### Step 1: Let the common root be \( \alpha \).
We assume that there is a common root \( \alpha \) that satisfies both equations. Thus, we have:
1. \( 3\alpha^2 + 4m\alpha + 2 = 0 \) (Equation 1)
2. \( 2\alpha^2 + 3\alpha - 2 = 0 \) (Equation 2)

### Step 2: Express \( \alpha^2 \) from Equation 2.
From Equation 2, we can express \( \alpha^2 \):
\[ 2\alpha^2 + 3\alpha - 2 = 0 \implies 2\alpha^2 = -3\alpha + 2 \implies \alpha^2 = -\frac{3}{2}\alpha + 1 \]

### Step 3: Substitute \( \alpha^2 \) into Equation 1.
Now, substitute \( \alpha^2 \) into Equation 1:
\[ 3\left(-\frac{3}{2}\alpha + 1\right) + 4m\alpha + 2 = 0 \]
Expanding this gives:
\[ -\frac{9}{2}\alpha + 3 + 4m\alpha + 2 = 0 \]
Combining like terms:
\[ \left(-\frac{9}{2} + 4m\right)\alpha + 5 = 0 \]

### Step 4: Set the coefficients to zero.
For the equation to hold for any \( \alpha \), both coefficients must be zero:
1. \( -\frac{9}{2} + 4m = 0 \)
2. \( 5 = 0 \) (which is not possible)

Thus, we only need to solve the first equation:
\[ -\frac{9}{2} + 4m = 0 \]
Solving for \( m \):
\[ 4m = \frac{9}{2} \implies m = \frac{9}{8} \]

### Step 5: Check for the second case.
Now, we also need to check if the second equation can yield another value for \( m \). We can use the condition for common roots:
Using the resultant method, we can set up the determinant condition:
\[ C_1A_2 - C_2A_1 = 0 \]
Where:
- \( A_1 = 3, B_1 = 4m, C_1 = 2 \)
- \( A_2 = 2, B_2 = 3, C_2 = -2 \)

Calculating:
\[ C_1A_2 - C_2A_1 = 2 \cdot 2 - (-2) \cdot 3 = 4 + 6 = 10 \]
Now, we can use the determinant condition:
\[ B_1C_2 - B_2C_1 = 4m \cdot (-2) - 3 \cdot 2 = -8m - 6 \]
And,
\[ A_1B_2 - A_2B_1 = 3 \cdot 3 - 2 \cdot 4m = 9 - 8m \]

Setting the resultant condition:
\[ 10^2 = (-8m - 6)(9 - 8m) \]
Expanding and simplifying will yield the quadratic equation in \( m \).

### Final Values of \( m \):
After solving the quadratic equation obtained from the resultant condition, we will find two values:
1. \( m = \frac{7}{4} \)
2. \( m = -\frac{11}{8} \)

### Conclusion:
The values of \( m \) for which the equations have a common root are:
\[ m = \frac{7}{4} \quad \text{and} \quad m = -\frac{11}{8} \]