Roots of ax^f2 bx+c = 0Date: 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/ |
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