If both the roots of the equation `x^(2) + bx + c = 0` lie in the interval (0,1) , then

Options

A. `b = - 1, c = 2`
B. `b gt - 2, c lt 1`
C. `b = - 5, c lt 2`
D. none of these

Answer

To solve the problem, we need to analyze the conditions under which both roots of the quadratic equation \(x^2 + bx + c = 0\) lie in the interval (0, 1).

### Step-by-Step Solution:

1. **Understanding the Roots**:
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). Since both roots lie in the interval (0, 1), we have:
\[ 0 < \alpha < 1 \quad \text{and} \quad 0 < \beta < 1 \]

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

3. **Analyzing the Sum of Roots**:
Since both roots are positive and less than 1:
\[ 0 < \alpha + \beta < 2 \]
This implies:
\[ 0 < -b < 2 \quad \Rightarrow \quad -2 < b < 0 \]

4. **Analyzing the Product of Roots**:
The product of the roots must also be positive and less than 1:
\[ 0 < \alpha \beta < 1 \]
This implies:
\[ 0 < c < 1 \]

5. **Conclusion**:
From the analysis, we conclude:
- \( c > 0 \) and \( c < 1 \) (i.e., \( 0 < c < 1 \))
- \( -2 < b < 0 \)

### Summary of Conditions:
- \( c > 0 \) and \( c < 1 \) (which means \( c \) is positive and less than 1)
- \( -2 < b < 0 \) (which means \( b \) is negative and greater than -2)