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. `2/3`
C. `-2/3`
D. `-1/3`

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 the coefficients
The given polynomial is in the standard form \( ax^2 + bx + c \), where:
- \( a = m \)
- \( b = 2 \)
- \( c = 3m \)

### Step 2: Use the formulas for the sum and product of the 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 coefficients identified:
\[ \text{Sum of the roots} = \alpha + \beta = -\frac{b}{a} = -\frac{2}{m} \]

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

### Step 5: Set up the equation
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 multiply both sides by \( m \) (assuming \( m \neq 0 \)):
\[ -2 = 3m \]
Now, divide both sides by 3:
\[ m = -\frac{2}{3} \]

### Conclusion
The value of \( m \) is \( -\frac{2}{3} \).

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