Question

Prove that ax^2+bx+c=0 has no rational roots if a,b,c are odd numbers

Answer 1

explanation:

if ax² + bx + c = 0 has rational roots then ax² + bx + c must factor across the integers. (that may need to be proven depending on your class - so write a lemma if necessary)

for ax² + bx + c to factor across the integers, there must be two integers that have a product of a∙c and a sum of b. (this also might need to be proven for your class level - again, write a lemma if necessary)

assuming the above two statements are true, consider a∙c.

since a and c are odd, a∙c is odd. Moreover, the only way a product of two integers can be odd is if both integers are odd.

now, the sum of any two odd integers is even. Since the product of a∙c is odd, it follows that any two integer factors whose product of a∙c will have a sum that is even. But since b is also odd, this can not happen.

thus, ax² + bx + c can not be factored across the integers if each of a, b, and c is odd. This implies that if each of a, b, and c is odd, ax² + bx + c = 0 will have no rational roots.

the square root of an odd number can be an integer. Consider 1, 9, 25, 49, 81, 121, etc.