If the roots of an equation `ax^(2) + bx + c=0` are positive , then which one of the following is correct ?

Options

A. Signs of a and c should be like
B. Signs of b and c should be like
C. Signs of a and b should be like
D. None of these

Explanation

To determine the conditions under which the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are positive, we can analyze the relationships between the coefficients \( a \), \( b \), and \( c \) based on the properties of the roots.

### Step-by-Step Solution:

1. **Understanding the Roots**:
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

2. **Condition for Positive Roots**:
For both roots \( \alpha \) and \( \beta \) to be positive:
- The sum of the roots \( \alpha + \beta \) must be positive.
- The product of the roots \( \alpha \beta \) must also be positive.

3. **Analyzing the Sum of Roots**:
From the sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} > 0 \]
This implies that \( -b \) and \( a \) must have the same sign. Therefore:
- If \( a > 0 \), then \( b < 0 \).
- If \( a < 0 \), then \( b > 0 \).

4. **Analyzing the Product of Roots**:
From the product of the roots:
\[ \alpha \beta = \frac{c}{a} > 0 \]
This implies that \( c \) and \( a \) must have the same sign. Therefore:
- If \( a > 0 \), then \( c > 0 \).
- If \( a < 0 \), then \( c < 0 \).

5. **Conclusion**:
For the roots \( \alpha \) and \( \beta \) to be positive:
- \( a \) and \( c \) must have the same sign.
- \( b \) must be negative if \( a > 0 \).

Thus, the correct option is that the signs of \( a \) and \( c \) should be the same.