If the roots are real and rational, theh Which of the following is the most appropriate one?

Options

A. `Dgt0`
B. `Dlt0`
C. D = 0
D. D is a perfect square

Answer

To solve the question regarding the conditions under which the roots of a quadratic equation are real and rational, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Quadratic Equation**:
A quadratic equation is generally expressed in the form:
\[ ax^2 + bx + c = 0 \]
where \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\).

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

3. **Conditions for Real and Rational Roots**:
- For the roots to be **real**, the discriminant must be **greater than or equal to zero**:
\[ D \geq 0 \]
- For the roots to be **rational**, the discriminant must be a **perfect square**. This means that \(D\) should be greater than zero and also a perfect square:
\[ D > 0 \quad \text{and} \quad D = k^2 \quad \text{for some integer } k \]

4. **Conclusion**:
Therefore, the most appropriate condition for the roots to be both real and rational is:
\[ D > 0 \]
This means the discriminant must be positive, ensuring that the roots are distinct and rational.

### Final Answer:
The most appropriate condition is \(D > 0\).

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