perplexus.info :: Just Math : Non-rational Quadratic

Non-rational Quadratic (Posted on 2010-08-22)

Prove that the quadratic equation ax² + bx + c = 0
does not have a rational root if a, b, and c are odd integers.

Submitted by Bractals

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Solution: (Hide)

Assume that p/q is a rational root of the equation with (p,q) = 1.
Then, ap² + bpq + cq² = 0. There are only three possible cases
for the integers p and q: both are odd or one is odd and the other
is even. In each of the three cases the sum of the three terms ap²,
bpq, and cq² is odd. This contradicts the assumption that the sum
should be even. Therefore, the equation has no rational root.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(2): Mod 4 .....back to 8	Ady TZIDON	2010-08-24 12:24:52
re: Mod 4	Steve Herman	2010-08-23 13:32:10
answer	Dej Mar	2010-08-23 04:09:24
Mod 4	Ady TZIDON	2010-08-23 02:38:00
By the numbers (spoiler)	Steve Herman	2010-08-22 21:21:50

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