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Topic: An odd quadratic (Read 321 times) |
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NickH
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An odd quadratic
« on: Sep 1st, 2003, 12:50pm » |
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If a, b, c are odd integers, show that the equation ax2 + bx + c = 0 has irrational roots.
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towr
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Re: An odd quadratic
« Reply #1 on: Sep 1st, 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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| « Last Edit: Sep 1st, 2003, 2:44pm by towr » |
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