For how many ordered triples of positive real numbers (a,b,c), do the equations ax^2+2bx+c, bx^2+2cx+a and cx^2+2ax+b each have 2 distinct real roots?

Answer 1

we want the discriminants to be positive. So,

b^2 - 4ac > 0
c^2 - 4ab > 0
a^2 - 4bc > 0

After playing around with this for a while, I was still stumped, so I went to wolframalpha.com, and it said that there were no all-positive solutions.

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