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Non-rational Quadratic (Posted on 2010-08-22) Difficulty: 2 of 5
Prove that the quadratic equation ax2 + bx + c = 0
does not have a rational root if a, b, and c are odd integers.

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Solution By the numbers (spoiler) | Comment 1 of 5
The quadratic has a rational root only if sqrt(b*b - 4ac) is rational.

Since (b*b - 4ac) is an integer, its square root can be rational only if (b*b - 4ac) is a perfect square.

Consider mod 8.

Since b is odd, b*b = 1 (mod 8)
Since a and c are odd, 4ac = 4 (mod 8)
Therefore, (b*b - 4ac) = 5 (mod 8)

But all perfect squares = 0 or 1 or 4 (mod 8). 

Therefore,  (b*b - 4ac) is not a perfect square, so its square root is not rational and the quadratic has no rational roots.

  Posted by Steve Herman on 2010-08-22 21:21:50
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