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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
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Associated Topics:
High School Number Theory

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