Final answer:
The quadratic equation ax² + bx + c = 0 with odd coefficients a, b, and c cannot have integer solutions since the discriminant produces an odd number, whose square root is non-integer, and an even denominator multiplies this non-integer solution.
Step-by-step explanation:
When a, b, and c are odd numbers, the quadratic equation ax² + bx + c = 0 cannot have integer solutions. We use the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), to solve for the roots of a quadratic equation. The discriminant part of this formula is b² - 4ac. In our case, since a, b, and c are odd, b² is an odd squared (which is odd), and 4ac is even times odd (which is even), their difference must be odd. Thus, the square root of an odd number is not an integer, leading to non-integer solutions for x.
Moreover, the denominator of the quadratic formula is 2a, and since a is odd, the denominator is even. Any non-integer numerator, when divided by an even number, cannot result in an integer. Therefore, the quadratic equation when a, b, and c are odd cannot yield integer solutions.