If the roots of a quadratic equation are real and rational, which one of the following best describes the value of the discriminant D ?

Options

A. `Dgt0`
B. `D=0`
C. `Dlt0`
D. `Dge0`

Answer

To determine the value of the discriminant \( D \) for a quadratic equation whose roots are real and rational, we can analyze the relationship between the discriminant and the nature of the roots.

### Step-by-Step Solution:

1. **Understanding the Discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[ D = b^2 - 4ac \]
The value of \( D \) helps us determine the nature of the roots of the quadratic equation.

2. **Conditions for the Roots**:
- If \( D < 0 \): The roots are complex (imaginary) and not real.
- If \( D = 0 \): The roots are real and equal (also known as repeated roots).
- If \( D > 0 \): The roots can be either real and unequal or real and rational, depending on whether \( D \) is a perfect square.

3. **Real and Rational Roots**:
- For the roots to be real and rational, \( D \) must be greater than 0 and also a perfect square. This ensures that the roots are not only real but also rational numbers.
- If \( D \) is greater than 0 but not a perfect square, the roots will be real and unequal but irrational.

4. **Conclusion**:
Since the question specifies that the roots are real and rational, the most suitable condition for the discriminant is:
\[ D > 0 \text{ and } D \text{ is a perfect square.} \]
Therefore, the best description of the value of the discriminant \( D \) is:
\[ D > 0 \]

### Final Answer:
The value of the discriminant \( D \) must be greater than 0.

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