# Problem 33

Algebra Level 5

$\large \dfrac{5a^2-3ab+2}{a^2b-a^3}$

Let $$a$$ and $$b$$ be non-zero real numbers such that all the roots of the equation $$ax^3-x^2+bx-1=0$$ are positive reals.

Given that the minimum value of the expression above can be expressed as $$H\sqrt{K}$$ and the equality achieved when $a=\dfrac{\sqrt{A}}{M}, \qquad b=\sqrt{S}$ where $$H,K,A,M$$ and $$S$$ are positive integers with $$K,A,S$$ are square-free number and $$\gcd(A,M)=1$$.

Compute $$H+K+A+M+S$$.

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