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

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Discussion Overview

The discussion revolves around finding the minimum value of the expression $\dfrac{|b|+|c|}{a}$, where $a$ is a positive real number and $\alpha, \beta, \gamma$ are the roots of the cubic equation $x^3 + ax + 1 = 0$. The roots of another cubic equation involving the ratios of these roots are also considered.

Discussion Character

  • Exploratory, Debate/contested, Mathematical reasoning

Main Points Raised

  • Post 1 introduces the problem and asks for the minimum value of $\dfrac{|b|+|c|}{a}$.
  • Post 2 suggests that the answer might be $3/a$.
  • Post 3 reiterates the suggestion of $3/a$ as the answer.
  • Post 4 questions whether the answer is a specific number.
  • Post 5 provides a different answer of $3888^{\tiny\dfrac{1}{6}}$, indicating a potential disagreement with previous suggestions.

Areas of Agreement / Disagreement

There is no consensus on the minimum value of $\dfrac{|b|+|c|}{a}$, with participants proposing different answers and questioning each other's suggestions.

Contextual Notes

The discussion includes varying interpretations of the problem and different proposed solutions, but lacks a clear resolution or agreement on the correct answer.

anemone
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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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