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Coefficients of Polynomial (Tomato subjective 69)

Problem: Suppose that the three equations \({\displaystyle{ax^2}} \) – \({\displaystyle {2bx + c = 0}} \) , \({\displaystyle{bx^2}} \) – \({\displaystyle {2cx + a = 0}} \) and \({\displaystyle{cx^2}} \) – \({\displaystyle {2ax + b = 0}} \) all have only positive roots. Show that a =b = c.
Solution: If possible let a, b, c are not all equal. \({\displaystyle{ax^2}} \) – \({\displaystyle {2bx + c = 0}} \) , \({\displaystyle{bx^2}} \) – \({\displaystyle {2cx + a = 0}} \) , \({\displaystyle{cx^2}} \) – \({\displaystyle {2ax + b = 0}} \) all have only positive roots.
So all of a, b, c cannot be its same sign as discriminant > 0 for three equations ( \({\displaystyle{b^2}} \) > $latex

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September 24, 2015

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