If D = 4477, then the roots are

Options

A. equal and real
B. irrational
C. conjugate pairs
D. rational and unequal

Answer

To solve the problem, we need to analyze the discriminant \( D \) given as \( 4477 \) and determine the nature of the roots of the corresponding quadratic equation.

### 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 nature of the roots depends on the value of \( D \):
- If \( D > 0 \): The roots are real and distinct.
- If \( D = 0 \): The roots are real and equal.
- If \( D < 0 \): The roots are complex (not real).

2. **Given Value of D**:
Here, we have \( D = 4477 \). Since \( 4477 > 0 \), we can conclude that the roots are real and distinct.

3. **Checking for Rationality**:
To determine if the roots are rational or irrational, we need to check if \( D \) is a perfect square. A perfect square is an integer that is the square of another integer.

4. **Finding the Square Root**:
We can find the square root of \( 4477 \):
\[ \sqrt{4477} \approx 66.8 \]
Since \( 66.8 \) is not an integer, \( 4477 \) is not a perfect square.

5. **Conclusion on the Nature of Roots**:
Since \( D > 0 \) and \( D \) is not a perfect square, the roots are real, distinct, and irrational.

### Final Answer:
The roots are **real, distinct, and irrational**.