In quadratic equation `ax^(2)+bx+c=0`, if discriminant `D=b^(2)-4ac`, then roots of quadratic equation are:

Options

A. real and distinct, if `D gt 0`
B. real and equal (repeated roots), if `D=0`
C. non-real (imaginary), if `D lt 0`
D. None of the above

Answer

To solve the problem regarding the roots of the quadratic equation \( ax^2 + bx + c = 0 \) based on the discriminant \( D = b^2 - 4ac \), we can analyze the conditions of the discriminant and how they relate to the nature of the roots.

### Step-by-Step Solution:

1. **Understanding the Discriminant**:
The discriminant \( D \) is given by the formula \( D = b^2 - 4ac \). It helps us determine the nature of the roots of the quadratic equation.

2. **Case 1: \( D > 0 \)**:
- If the discriminant \( D \) is greater than zero, it indicates that the quadratic equation has two distinct real roots.
- The roots can be calculated using the quadratic formula:
\[ x_1 = \frac{-b + \sqrt{D}}{2a}, \quad x_2 = \frac{-b - \sqrt{D}}{2a} \]

3. **Case 2: \( D = 0 \)**:
- If the discriminant \( D \) is equal to zero, it indicates that the quadratic equation has exactly one real root, which is a repeated root.
- The root can be calculated as:
\[ x = \frac{-b}{2a} \]

4. **Case 3: \( D < 0 \)**:
- If the discriminant \( D \) is less than zero, it indicates that the quadratic equation has no real roots; instead, it has two complex (imaginary) roots.
- The roots can be expressed as:
\[ x_1 = \frac{-b + i\sqrt{|D|}}{2a}, \quad x_2 = \frac{-b - i\sqrt{|D|}}{2a} \]
where \( i \) is the imaginary unit.

5. **Conclusion**:
Based on the value of the discriminant \( D \):
- If \( D > 0 \): Roots are real and distinct.
- If \( D = 0 \): Roots are real and equal (repeated).
- If \( D < 0 \): Roots are non-real (imaginary).

### Final Answer:
The roots of the quadratic equation can be summarized as follows:
- Real and distinct if \( D > 0 \)
- Real and equal if \( D = 0 \)
- Non-real (imaginary) if \( D < 0 \)