If a, b, c are odd integers, then the roots of `ax^(2)+bx+c=0`, if real, cannot be

Options

A. integers
B. rational numbers
C. irrational
D. equal

Answer

To solve the problem, we need to analyze the quadratic equation \( ax^2 + bx + c = 0 \) where \( a, b, c \) are odd integers. We want to determine what type of roots this equation can have if they are real.

### Step 1: Understanding the condition for real roots
The roots of a quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the roots to be real, the discriminant must be non-negative:
\[ D = b^2 - 4ac \geq 0 \]

**Hint:** Remember that the discriminant determines the nature of the roots. If \( D < 0 \), the roots are complex; if \( D = 0 \), the roots are real and equal; if \( D > 0 \), the roots are real and distinct.

### Step 2: Analyzing the discriminant
Given that \( a, b, c \) are odd integers, 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:
\[ D = (2n + 1)^2 - 4(2m + 1)(2p + 1) \]
Calculating \( D \):
\[ D = 4n^2 + 4n + 1 - 4[(2m)(2p) + 2m + 2p + 1] \]
\[ D = 4n^2 + 4n + 1 - 16mp - 8m - 8p - 4 \]
\[ D = 4n^2 + 4n - 16mp - 8m - 8p - 3 \]

**Hint:** Simplifying the discriminant helps to see if it can be non-negative.

### Step 3: Analyzing the parity of \( D \)
Notice that \( D \) is an expression that is dependent on \( n, m, p \). The key observation is that:
- The term \( 4n^2 + 4n \) is always even.
- The term \( -16mp - 8m - 8p \) is also even.
- The constant term \( -3 \) is odd.

Thus, the entire expression for \( D \) is even minus odd, which results in an odd number:
\[ D \text{ is odd} \]

**Hint:** The parity (even or odd nature) of the discriminant is crucial in determining the nature of the roots.

### Step 4: Conclusion about the roots
Since \( D \) is odd, it cannot be zero. Therefore, the roots cannot be equal. Furthermore, since \( D \) is positive, the roots are real but they cannot be rational because the square root of an odd number is irrational.

Thus, we conclude that if \( a, b, c \) are odd integers, the roots of the equation \( ax^2 + bx + c = 0 \) cannot be rational numbers, integers, or equal.

### Final Answer
The roots of \( ax^2 + bx + c = 0 \) cannot be:
- Rational numbers
- Integers
- Equal