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

Options

A. `(1)/(3)`
B. `(-1)/(3)`
C. `(2)/(3)`
D. `(-2)/(3)`

Explanation

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 the product of the roots.

### Step-by-Step Solution:

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

2. **Use the formulas for the sum and product of the roots**:
- The sum of the roots \( (\alpha + \beta) \) is given by the formula:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{2}{k} \]
- The product of the roots \( (\alpha \beta) \) is given by the formula:
\[ \alpha \beta = \frac{c}{a} = \frac{3k}{k} = 3 \]

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

4. **Solve for \( k \)**:
To solve for \( k \), we can multiply both sides by \( k \) (assuming \( k \neq 0 \)):
\[ -2 = 3k \]
Now, divide both sides by 3:
\[ k = -\frac{2}{3} \]

5. **Conclusion**:
The value of \( k \) is \( -\frac{2}{3} \).

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

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