A quadratic equation `ax^2+bx+c=0` may have

Options

A. two real roots
B. two non-real roots
C. one real and one non-real root
D. Either (a) or (b)

Answer

To determine the nature of the roots of a quadratic equation of the form \( ax^2 + bx + c = 0 \), we can analyze the discriminant, denoted as \( D \), which is calculated using the formula:

\[ D = b^2 - 4ac \]

### Step-by-Step Solution:

1. **Identify the Discriminant**:
The discriminant \( D \) is given by the expression \( b^2 - 4ac \).

**Hint**: Remember that the discriminant helps us understand the nature of the roots of the quadratic equation.

2. **Analyze the Discriminant**:
- **Case 1**: If \( D = 0 \):
- The roots are real and equal.
- This means there is one unique solution (a repeated root).

**Hint**: When the discriminant is zero, it indicates that the parabola touches the x-axis at one point.

- **Case 2**: If \( D < 0 \):
- The roots are non-real (complex) and conjugate pairs.
- This means there are no real solutions.

**Hint**: A negative discriminant indicates that the parabola does not intersect the x-axis.

- **Case 3**: If \( D > 0 \):
- The roots are real and distinct.
- This means there are two different solutions.

**Hint**: A positive discriminant indicates that the parabola intersects the x-axis at two points.

3. **Conclusion**:
Based on the analysis of the discriminant, we can conclude that:
- A quadratic equation can have:
- Two real and distinct roots (when \( D > 0 \)),
- One real and equal root (when \( D = 0 \)),
- Two non-real roots (when \( D < 0 \)).

Therefore, the possible types of roots for a quadratic equation are:
- Two real roots
- One real root (repeated)
- Two non-real roots

**Final Answer**: A quadratic equation \( ax^2 + bx + c = 0 \) may have two real roots, one real root (repeated), or two non-real roots.