If the roots are unequal, then the possible values of D is/are

Options

A. D = 0
B. `D-{0}`
C. `Dgt0`
D. `Dlt0`

Answer

To determine the possible values of D (the discriminant) for the roots of a quadratic equation to be unequal, we can follow these steps:

### Step 1: Understand the Discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[ D = b^2 - 4ac \]
The nature of the roots of the quadratic equation depends on the value of \( D \).

### Step 2: Analyze the Conditions for Roots
1. **If \( D > 0 \)**: The quadratic equation has two distinct (unequal) real roots.
2. **If \( D = 0 \)**: The quadratic equation has exactly one real root (a repeated root).
3. **If \( D < 0 \)**: The quadratic equation has two complex (imaginary) roots, which are also unequal.

### Step 3: Identify the Values of D for Unequal Roots
From the analysis:
- For the roots to be unequal, \( D \) must be either greater than 0 or less than 0.
- Therefore, the possible values of \( D \) are:
- \( D > 0 \) (two distinct real roots)
- \( D < 0 \) (two distinct complex roots)

### Conclusion
The possible values of \( D \) for the roots to be unequal are:
- \( D > 0 \) or \( D < 0 \)

### Final Answer
Thus, the correct options are:
- \( D \) is positive (greater than 0)
- \( D \) is negative (less than 0)