If the graph a quadratic polynomial lies entirely above x-axis. Which one of the following is correct ?

Options

A. Both the roots are real
B. One root is real and the other is complex
C. Both the roots are complex
D. Can not say

Answer

To solve the question, we need to analyze the properties of the graph of a quadratic polynomial and its roots when the graph lies entirely above the x-axis.

### Step-by-Step Solution:

1. **Understanding the Quadratic Polynomial**:
A quadratic polynomial can be expressed in the standard form as:
\[ f(x) = Ax^2 + Bx + C \]
where \(A\), \(B\), and \(C\) are constants and \(A \neq 0\).

2. **Graph Behavior**:
If the graph of this quadratic polynomial lies entirely above the x-axis, it means that for all values of \(x\), \(f(x) > 0\). This indicates that there are no points where the graph intersects or touches the x-axis.

3. **Roots of the Quadratic**:
The roots of the quadratic equation \(Ax^2 + Bx + C = 0\) can be determined using the discriminant \(D\):
\[ D = B^2 - 4AC \]
The nature of the roots is determined by the value of \(D\):
- If \(D > 0\), there are two distinct real roots.
- If \(D = 0\), there is one real root (a repeated root).
- If \(D < 0\), there are two complex roots.

4. **Condition for Entirely Above the x-axis**:
Since the graph lies entirely above the x-axis, it cannot have any real roots (as real roots would imply the graph touches or crosses the x-axis). Therefore, the discriminant must be less than zero:
\[ D < 0 \implies B^2 - 4AC < 0 \]
This indicates that the quadratic has two complex roots.

5. **Conclusion**:
Based on the analysis, the correct statement is that both roots are complex. Therefore, the answer is:
**Option C: Both the roots are complex.**