If the comprehensive graph of quadratic function lies in all the.four quadrants, then roots are necessarily

Options

A. real
B. non-real
C. real and equal
D. equal and irrational

Answer

To solve the problem, we need to analyze the conditions under which a quadratic function can have its graph lie in all four quadrants of the Cartesian plane.

### Step-by-Step Solution:

1. **Understanding the Quadratic Function**:
A quadratic function is generally expressed in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The graph of this function is a parabola.

**Hint**: Remember that the shape of the parabola depends on the coefficient \( a \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

2. **Graph Behavior**:
For the graph of the quadratic function to lie in all four quadrants, it must extend infinitely in both the positive and negative directions on the y-axis. This means that the parabola must cross the x-axis at two points (real roots).

**Hint**: Think about how the parabola must behave at extreme values of \( x \) to ensure it reaches both positive and negative y-values.

3. **Roots of the Quadratic Function**:
The roots of the quadratic function can be determined using the discriminant \( D = b^2 - 4ac \).
- If \( D > 0 \), the quadratic has two distinct real roots.
- If \( D = 0 \), the quadratic has one real root (a repeated root).
- If \( D < 0 \), the quadratic has no real roots (the roots are complex or non-real).

**Hint**: Consider how the discriminant affects the number and type of roots.

4. **Conclusion**:
Since the graph of the quadratic function must cross the x-axis to lie in all four quadrants, it must have two distinct real roots. Therefore, the roots of the quadratic function are necessarily real.

**Final Answer**: The roots are necessarily real.