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        riddles >> easy >> An odd quadratic
        
(Message started by: NickH on Sep 1^st, 2003, 12:50pm)

Title: An odd quadratic
Post by NickH on Sep 1^st, 2003, 12:50pm

If a, b, c are odd integers, show that the equation ax² + bx + c = 0 has irrational roots.

Title: Re: An odd quadratic
Post by towr on Sep 1^st, 2003, 2:42pm

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D = (b²-4ac) {needs to be a square, if we want (b+-[sqrt]D)/a to be rational}
= (b - 2d)² {an odd square, since b is odd and 4ac even (d is some integer) }
= b² - 4bd + 4d²

4ac = 4bd - 4d²
ac = bd - d²
ac = d(b-d)

ac is odd since a and c are odd, but d(b+d) can obviously never be odd when b is odd.
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