TITLE: Solving Equation through inequalities.
QUESTION [12 upvotes]: If $x^6-12x^5+ax^4+bx^3+cx^2+dx+64=0$ has positive roots then find $a,b,c,d$.

I did something but that don't deserve to be added here, but what I thought before doing that is following:

For us, Product and Sum of roots are given.
Roots are positive.
Hence I tried to use AM-GM-HM inequalities, as sum and product are given, but I failed to conclude something good.

So please deliver some hints or solution regarding AM-GM-HM inequalities.

REPLY [6 votes]: Let $x_i$ be our roots.
Now, by AM-GM $$2=\frac{\sum\limits_{i=1}^6x_i}{6}\geq\sqrt[6]{\prod_{i=1}^6x_i}=2,$$
which says that all $x_i=2$.
Thus, $$a=2^2\cdot\frac{6(6-1)}{2}=60,$$ $$b=-2^3\cdot\frac{6(6-1)(6-2)}{6}=-160,$$
$$c=2^4\cdot\frac{6(6-1)(6-2)(6-3)}{24}=240$$ and
$$d=-2^5\cdot\frac{6(6-1)(6-2)(6-3)(6-4)}{120}=-192.$$