If `a gt 0` and both the roots of `ax^(2) + bx + c = 0` are more than 1, then

Options

A. `a + b + c gt 0`
B. `a + b + 4c = 0`
C. `a + b + c lt 0`
D. `a + 4b + c = 0`

Answer

To solve the problem, we need to analyze the conditions given in the question. We know that the quadratic equation is of the form:

\[ ax^2 + bx + c = 0 \]

where \( a > 0 \) and both roots (let's denote them as \( \alpha \) and \( \beta \)) are greater than 1.

### Step-by-step Solution:

1. **Understanding the Roots**:
Since both roots \( \alpha \) and \( \beta \) are greater than 1, we can express this mathematically as:
\[ \alpha > 1 \quad \text{and} \quad \beta > 1 \]

2. **Using Vieta's Formulas**:
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

3. **Analyzing the Sum of Roots**:
Since both roots are greater than 1:
\[ \alpha + \beta > 1 + 1 = 2 \]
Therefore, we have:
\[ -\frac{b}{a} > 2 \implies b < -2a \]

4. **Analyzing the Product of Roots**:
Similarly, since both roots are greater than 1:
\[ \alpha \beta > 1 \cdot 1 = 1 \]
Thus, we have:
\[ \frac{c}{a} > 1 \implies c > a \]

5. **Combining the Conditions**:
From the above analysis, we have derived two conditions:
- \( b < -2a \)
- \( c > a \)

6. **Conclusion**:
The conditions derived indicate that if \( a > 0 \) and both roots of the quadratic equation are greater than 1, then we can conclude that:
- \( A + B + C > 0 \) where \( A = a \), \( B = b \), and \( C = c \).

### Final Result:
Thus, the correct option based on the analysis is:
\[ A + B + C > 0 \]