Associated Topics || Dr. Math Home || Search Dr. Math

### Roots of ax^f2 bx+c = 0

```
Date: 05/22/97 at 14:09:18
From: Anonymous
Subject: Roots not rational

Prove that if a,b,c are odd integers, then the roots of ax^2 bx+c=0
are not rational.
```

```
Date: 05/23/97 at 19:49:55
From: Doctor Anthony
Subject: Re: Roots not rational

Suppose the roots are rational numbers in the form (p/q) and (r/s).
Then the equation could be written

(x - p/q)(x - r/s) = 0

x^2 - (p/q + r/s)x + (pr)/(qs) = 0

x^2 - [(ps+qr)/(qs)]x + (pr)/(qs) = 0

(qs)x^2 - (ps+qr)x + pr = 0

Comparing with ax^2 + bx + c = 0  we have  a = qs

b = -(ps+qr)

c = pr

Since a is odd, then both q and s are odd.  Since c is odd, then both
p and r are odd.  So all four of p, q, r, s are odd.

This means ps is odd and qr is odd, but the sum of two odd numbers is
even. That is, -(ps+qr) is even so b must be even. However, we are
told that b is odd. It follows that we could not have rational roots
in the form p/q, r/s.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search