To solve the problem, we need to analyze the quadratic equation given by \( ax^2 + bx + c = 0 \) where \( a, b, c \) are odd integers. We want to determine the nature of the roots of this equation when they are real.
### Step-by-Step Solution:
1. **Understanding the Quadratic Equation**:
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients.
2. **Identifying the Conditions**:
We know that \( a, b, c \) are odd integers. This means that \( a \equiv 1 \mod 2 \), \( b \equiv 1 \mod 2 \), and \( c \equiv 1 \mod 2 \).
3. **Finding the Discriminant**:
The roots of the quadratic equation can be found using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
The term under the square root, \( b^2 - 4ac \), is called the discriminant (\( D \)). For the roots to be real, \( D \) must be non-negative:
\[
D = b^2 - 4ac \geq 0
\]
4. **Calculating the Discriminant**:
Since \( a, b, c \) are odd, we can express them as:
- \( a = 2m + 1 \)
- \( b = 2n + 1 \)
- \( c = 2p + 1 \)
for some integers \( m, n, p \).
Now, substituting these into the discriminant:
\[
D = (2n + 1)^2 - 4(2m + 1)(2p + 1)
\]
Expanding this gives:
\[
D = 4n^2 + 4n + 1 - 4(4mp + 2m + 2p + 1)
\]
Simplifying further:
\[
D = 4n^2 + 4n + 1 - (16mp + 8m + 8p + 4)
\]
\[
D = 4n^2 + 4n - 16mp - 8m - 8p - 3
\]
5. **Analyzing the Result**:
Since \( D \) is a combination of even and odd terms, we need to see if it can be non-negative. However, notice that \( b^2 \) is odd (since \( b \) is odd) and \( 4ac \) is even (since \( a \) and \( c \) are odd, their product is odd, and multiplying by 4 gives an even number). Thus, \( D \) is odd minus even, which results in an odd number.
6. **Conclusion**:
Since \( D \) is odd, it cannot be zero or positive (as it must be non-negative for real roots). Therefore, if the roots are real, they cannot be rational (as rational roots would imply a rational discriminant).
### Final Answer:
Thus, the roots of \( ax^2 + bx + c = 0 \), if real, cannot be rational numbers.
