If the roots are real and equal.then which of the following is always valid?

Options

A. 1. `Dgt0`
B. 2. `Dlt0`
C. 3. D = 0
D. 4. D is a perfect square

Answer

To solve the question, we need to understand the condition for a quadratic equation to have real and equal roots.

### 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 of the Quadratic Equation**:
The discriminant \(D\) of the quadratic equation is given by:
\[ D = b^2 - 4ac \]
The nature of the roots of the quadratic equation depends on the value of the discriminant.

3. **Condition for Real and Equal Roots**:
For the roots to be real and equal, the discriminant must be equal to zero:
\[ D = 0 \implies b^2 - 4ac = 0 \]

4. **Identifying the Valid Statement**:
Given the options in the question, we need to identify which statement is always valid when the roots are real and equal. The correct option will state that:
\[ b^2 - 4ac = 0 \]
This means that the third option, which states that the discriminant \(D\) is equal to zero, is always valid.

5. **Conclusion**:
Therefore, if the roots of the quadratic equation are real and equal, the statement that is always valid is:
\[ b^2 - 4ac = 0 \]
Hence, the answer is the third option.