If the sum of the roots of `kx^(2) + 2x + 3k = 0` is equal to their product, then k =

Options

A. `(3)/(2)`
B. `(2)/(3)`
C. `(-2)/(3)`
D. `(-3)/(2)`

Answer

To solve the problem, we need to find the value of \( k \) such that the sum of the roots of the quadratic equation \( kx^2 + 2x + 3k = 0 \) is equal to their product.

### Step-by-step Solution:

1. **Identify the coefficients**:
The given quadratic equation is \( kx^2 + 2x + 3k = 0 \). Here, we can identify:
- \( a = k \)
- \( b = 2 \)
- \( c = 3k \)

2. **Use the formulas for sum and product of roots**:
For a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta \) is given by \( -\frac{b}{a} \).
- The product of the roots \( \alpha \beta \) is given by \( \frac{c}{a} \).

3. **Calculate the sum of the roots**:
\[ \text{Sum of the roots} = -\frac{b}{a} = -\frac{2}{k} \]

4. **Calculate the product of the roots**:
\[ \text{Product of the roots} = \frac{c}{a} = \frac{3k}{k} = 3 \]

5. **Set the sum equal to the product**:
According to the problem, the sum of the roots is equal to their product:
\[ -\frac{2}{k} = 3 \]

6. **Solve for \( k \)**:
To eliminate the fraction, multiply both sides by \( k \) (assuming \( k \neq 0 \)):
\[ -2 = 3k \]
Now, solve for \( k \):
\[ k = -\frac{2}{3} \]

### Final Answer:
Thus, the value of \( k \) is \( -\frac{2}{3} \).