Finding Min Value of $\dfrac{|b|+|c|}{a}$ from Roots of Cubic Equations

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The minimum value of $\dfrac{|b|+|c|}{a}$, given the roots $\alpha, \beta, \gamma$ of the cubic equation $x^3 + ax + 1 = 0$ and the ratios $\dfrac{\alpha}{\beta}, \dfrac{\beta}{\gamma}, \dfrac{\gamma}{\alpha}$ as roots of $x^3 + bx^2 + cx - 1 = 0$, is definitively $3888^{\tiny\dfrac{1}{6}}$. This conclusion was reached after evaluating the relationships between the coefficients and roots of the cubic equations involved.

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If $\alpha,\,\beta,\,\gamma$ are the roots of the equation $x^3+ax+1=0$, where $a$ is a positive real number and $\dfrac{\alpha}{\beta},\,\dfrac{\beta}{\gamma},\,\dfrac{\gamma}{\alpha}$ be the roots of the equation $x^3+bx^2+cx-1=0$, find the minimum value of $\dfrac{|b|+|c|}{a}$.
 
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is the answer 3/a
anemone said:
If $\alpha,\,\beta,\,\gamma$ are the roots of the equation $x^3+ax+1=0$, where $a$ is a positive real number and $\dfrac{\alpha}{\beta},\,\dfrac{\beta}{\gamma},\,\dfrac{\gamma}{\alpha}$ be the roots of the equation $x^3+bx^2+cx-1=0$, find the minimum value of $\dfrac{|b|+|c|}{a}$.

Is the answer $3/a$
 
solakis said:
Is the answer $3/a$

Nope, sorry solakis!
 
is the answer a number?
 
Hi solakis and to all MHB members,

I am sorry that I still haven't gotten around to follow up all my unanswered challenges here in MHB! (Sadface) I promise that once I got my personal things straighten out a bit more, I will have more time for MHB then.

To answer to your query, solakis, the answer to this problem is $3888^{\tiny\dfrac{1}{6}}$.
 

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