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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