The expression `y = ax^2+ bx + c` has always the same sign as of a if `(A) 4ac < b^2 (B) 4ac > b^2 (C) 4ac= b2 (D) ac < b^2`

Options

A. `4 ac lt b^(2)`
B. `4 ac gt b^(2)`
C. `ac lt b^(2)`
D. `ac gt b^(2)`

Answer

To determine when the expression \( y = ax^2 + bx + c \) has the same sign as \( a \), we need to analyze the conditions under which the quadratic expression does not change its sign.

### Step-by-Step Solution:

1. **Understanding the Quadratic Expression**:
The expression \( y = ax^2 + bx + c \) is a quadratic function. The sign of \( y \) depends on the coefficient \( a \) and the discriminant \( D \).

2. **Identifying the Conditions**:
- If \( a > 0 \): The parabola opens upwards. For \( y \) to always be positive, the quadratic must not intersect the x-axis, which occurs when the discriminant \( D < 0 \).
- If \( a < 0 \): The parabola opens downwards. For \( y \) to always be negative, the quadratic must also not intersect the x-axis, which again requires \( D < 0 \).

3. **Calculating the Discriminant**:
The discriminant \( D \) of the quadratic equation is given by:
\[ D = b^2 - 4ac \]
For the quadratic to not have real roots (and thus not change sign), we need:
\[ D < 0 \implies b^2 - 4ac < 0 \]
Rearranging gives:
\[ b^2 < 4ac \]

4. **Analyzing the Conditions**:
- If \( a > 0 \) and \( b^2 < 4ac \), then \( y \) is always positive.
- If \( a < 0 \) and \( b^2 < 4ac \), then \( y \) is always negative.

5. **Conclusion**:
The condition for \( y \) to always have the same sign as \( a \) is that the discriminant must be less than zero:
\[ 4ac > b^2 \]
Therefore, the correct option is \( (B) \, 4ac > b^2 \).