If the graph of `y=ax^(2)+bx+c` lies completely above the x-axis, then

Options

A. a) `b^(2) - 4ac gt 0, a gt 0`
B. b) `b^(2) - 4ac gt 0, a lt 0`
C. c) `b^(2) - 4ac lt 0, a gt 0`
D. d) `b^(2) - 4ac lt 0, a lt 0`

Answer

To solve the problem, we need to analyze the conditions under which the graph of the quadratic equation \(y = ax^2 + bx + c\) lies completely above the x-axis.

### Step-by-Step Solution:

1. **Understanding the Graph's Position**:
- For the graph of the quadratic function to lie completely above the x-axis, it must not intersect the x-axis at any point. This means that the quadratic equation must not have any real roots.

2. **Identifying the Roots**:
- The roots of the quadratic equation can be determined using the discriminant, which is given by the formula:
\[ D = b^2 - 4ac \]
- If the graph does not intersect the x-axis, the discriminant must be less than zero:
\[ D < 0 \quad \Rightarrow \quad b^2 - 4ac < 0 \]

3. **Condition on the Coefficient 'a'**:
- Additionally, for the parabola to open upwards (which is necessary for it to lie above the x-axis), the coefficient \(a\) must be positive:
\[ a > 0 \]

4. **Conclusion**:
- Therefore, for the graph of \(y = ax^2 + bx + c\) to lie completely above the x-axis, the following conditions must hold:
- \(b^2 - 4ac < 0\) (discriminant is negative)
- \(a > 0\) (the parabola opens upwards)

### Final Answer:
The conditions required for the graph of \(y = ax^2 + bx + c\) to lie completely above the x-axis are:
- \(b^2 - 4ac < 0\)
- \(a > 0\)