The expression `ax^(2)+bx+c` has the same sign as of a if

Options

A. `b^(2)-4ac gt 0`
B. `b^(2)-4ac= 0`
C. `b^(2)-4ac lt 0`
D. b and c have the same sign as of a

Answer

To determine the conditions under which the expression \( ax^2 + bx + c \) has the same sign as \( a \), we need to analyze the quadratic expression based on the value of \( a \) and the discriminant.

### Step-by-Step Solution:

1. **Identify the Quadratic Expression**:
We are given the quadratic expression \( ax^2 + bx + c \).

2. **Consider the Sign of \( a \)**:
- If \( a > 0 \): The parabola opens upwards.
- If \( a < 0 \): The parabola opens downwards.

3. **Determine the Conditions for Same Sign**:
- For \( ax^2 + bx + c \) to have the same sign as \( a \), we need to consider the roots of the quadratic equation \( ax^2 + bx + c = 0 \).
- The roots can be found using the discriminant \( D = b^2 - 4ac \).

4. **Analyze the Discriminant**:
- If \( D < 0 \): The quadratic does not intersect the x-axis and retains the same sign for all \( x \).
- If \( D \geq 0 \): The quadratic may change sign at the roots.

5. **Conclusion Based on the Value of \( a \)**:
- If \( a > 0 \) and \( D < 0 \): \( ax^2 + bx + c > 0 \) for all \( x \).
- If \( a < 0 \) and \( D < 0 \): \( ax^2 + bx + c < 0 \) for all \( x \).
- Therefore, the condition for \( ax^2 + bx + c \) to have the same sign as \( a \) is that the discriminant must be less than zero:
\[ b^2 - 4ac < 0 \]

### Final Answer:
The expression \( ax^2 + bx + c \) has the same sign as \( a \) if:
\[ b^2 - 4ac < 0 \]