If the roots `ax^(2)+bx+c=0` are both negative and `b lt 0`, then

Options

A. `a lt 0, c lt 0`
B. `a lt 0, c gt 0`
C. `a gt 0, c lt 0`
D. `a gt 0, c gt 0`

Answer

To solve the problem, we need to analyze the quadratic equation \( ax^2 + bx + c = 0 \) under the conditions that both roots are negative and \( b < 0 \).

### Step-by-Step Solution:

1. **Understanding the Roots**:
Let the roots of the equation be \( \alpha \) and \( \beta \). Since both roots are negative, we can denote them as \( \alpha < 0 \) and \( \beta < 0 \).

2. **Product of Roots**:
The product of the roots \( \alpha \) and \( \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} \]
Since both \( \alpha \) and \( \beta \) are negative, their product \( \alpha \beta \) is positive. Therefore:
\[ \frac{c}{a} > 0 \]
This implies that \( c \) and \( a \) must either both be positive or both be negative.

3. **Sum of Roots**:
The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
Since both roots are negative, their sum \( \alpha + \beta \) is also negative. Therefore:
\[ -\frac{b}{a} < 0 \]
This implies that \( \frac{b}{a} > 0 \).

4. **Analyzing the Signs**:
From the inequalities derived:
- \( \frac{c}{a} > 0 \) implies \( c \) and \( a \) have the same sign.
- \( \frac{b}{a} > 0 \) implies \( b \) and \( a \) have the same sign.

5. **Considering \( b < 0 \)**:
Since we are given that \( b < 0 \), it follows that \( a \) must also be negative (to maintain \( \frac{b}{a} > 0 \)). Thus:
\[ a < 0 \]

6. **Conclusion about \( c \)**:
Since \( a < 0 \) and \( \frac{c}{a} > 0 \), it follows that \( c \) must also be negative:
\[ c < 0 \]

### Final Result:
The conditions that must hold true are:
- \( a < 0 \)
- \( c < 0 \)

Thus, the correct condition is:
\[ \text{Option: } a < 0, c < 0 \]