SOLUTION: if a, b, and c in a quadratic equation are all integers, is the product always rational? explain

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Question 407617: if a, b, and c in a quadratic equation are all integers, is the product always rational? explain
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
The product of the roots is always rational. Suppose that the quadratic is ax%5E2+%2B+bx+%2B+c+=+0. I'll divide both sides by a to obtain x%5E2+%2B+%28b%2Fa%29x+%2B+%28c%2Fa%29+=+0. If we assume the roots are r%5B1%5D and r%5B2%5D, then

%28x-r%5B1%5D%29%28x-r%5B2%5D%29+=+x%5E2+-+%28r%5B1%5D+%2B+r%5B2%5D%29x+%2B+r%5B1%5Dr%5B2%5D --> r%5B1%5Dr%5B2%5D+=+c%2Fa and r%5B1%5D+%2B+r%5B2%5D+=+-b%2Fa. These are sometimes called Vieta's formulas.