If the sum of the zeroes of the quadratic polynomial   f(x) = `mx^(2) + 2x + 3m`   is equal to their product, then m equals

Options

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

Answer

To solve the problem, we need to find the value of \( m \) such that the sum of the zeroes of the quadratic polynomial \( f(x) = mx^2 + 2x + 3m \) is equal to their product.

### Step 1: Identify coefficients
The given polynomial is \( f(x) = mx^2 + 2x + 3m \). We can identify the coefficients as follows:
- \( a = m \)
- \( b = 2 \)
- \( c = 3m \)

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

### Step 3: Calculate the sum of the roots
Using the formula for the sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{2}{m} \]

### Step 4: Calculate the product of the roots
Using the formula for the product of the roots:
\[ \alpha \beta = \frac{c}{a} = \frac{3m}{m} = 3 \]

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

### Step 6: Solve for \( m \)
To solve for \( m \), we can cross-multiply:
\[ -2 = 3m \]
Now, divide both sides by 3:
\[ m = -\frac{2}{3} \]

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