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 question regarding the roots of the quadratic equation \( ax^2 + bx + c = 0 \) based on the discriminant \( D = b^2 - 4ac \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Quadratic Equation**: The standard form of a quadratic equation is given as:
\[ ax^2 + bx + c = 0 \]

2. **Define the Discriminant**: The discriminant \( D \) is defined as:
\[ D = b^2 - 4ac \]

3. **Determine the Nature of Roots Based on the Discriminant**:
- If \( D > 0 \): The roots are **real and distinct**. This means there are two different real solutions for the equation.
- If \( D = 0 \): The roots are **real and equal** (or repeated). This means there is one real solution that occurs twice.
- If \( D < 0 \): The roots are **non-real** (or imaginary). This means there are no real solutions, and the solutions are complex numbers.

4. **Conclusion**: Based on the value of the discriminant \( D \), we can summarize the nature of the roots:
- **Real and Distinct**: \( D > 0 \)
- **Real and Equal**: \( D = 0 \)
- **Non-Real (Imaginary)**: \( D < 0 \)

### Final Answer:
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) depend on the discriminant \( D \) as follows:
- **Real and Distinct** if \( D > 0 \)
- **Real and Equal** if \( D = 0 \)
- **Non-Real (Imaginary)** if \( D < 0 \)

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