If the roots of `ax^(2) + bx + c = 0 (a gt 0)` be each greater than unity, then

Options

A. a + b + c = 0
B. a + b + c `gt` 0
C. a + b + c `lt` 0
D. none of these

Explanation

To solve the problem, we need to establish the conditions under which both roots of the quadratic equation \( ax^2 + bx + c = 0 \) (where \( a > 0 \)) are greater than unity.

### Step-by-Step Solution:

1. **Understanding the Roots**:
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). According to the problem, both roots are greater than 1, i.e., \( \alpha > 1 \) and \( \beta > 1 \).

2. **Using Vieta's Formulas**:
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

3. **Condition for Roots Greater than 1**:
Since both roots are greater than 1, we can derive the following conditions:
- For the sum of the roots:
\[ \alpha + \beta > 2 \implies -\frac{b}{a} > 2 \implies b < -2a \]
- For the product of the roots:
\[ \alpha \beta > 1 \implies \frac{c}{a} > 1 \implies c > a \]

4. **Evaluating the Function at \( x = 1 \)**:
To ensure that the quadratic does not cross the x-axis between the roots (since both roots are greater than 1), we evaluate the quadratic function at \( x = 1 \):
\[ f(1) = a(1)^2 + b(1) + c = a + b + c \]
Since both roots are greater than 1, we need \( f(1) > 0 \):
\[ a + b + c > 0 \]

5. **Summary of Conditions**:
Therefore, the conditions for the quadratic equation \( ax^2 + bx + c = 0 \) to have both roots greater than 1 are:
- \( b < -2a \)
- \( c > a \)
- \( a + b + c > 0 \)

### Conclusion:
The conditions that must be satisfied for the roots of the quadratic equation \( ax^2 + bx + c = 0 \) to be each greater than unity are:
1. \( b < -2a \)
2. \( c > a \)
3. \( a + b + c > 0 \)