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

• MHB
• anemone
In summary, the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation is important in understanding the behavior and possible range of values of the equation. It can be found by solving the equation and substituting the roots, and can be positive, negative, or zero. This value can also be used to determine the nature of the roots and has practical applications in various fields.
anemone
Gold Member
MHB
POTW Director
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 $\dfrac{|b|+|c|}{a}$.

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!

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 $3888^{\tiny\dfrac{1}{6}}$.

## 1. What is the purpose of finding the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation?

The minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation can be used to determine the minimum distance between the x-axis and the curve of the cubic equation. This can be useful in various applications, such as optimization problems in engineering and economics.

## 2. How do you find the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation?

To find the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation, you can first use the cubic formula to find the roots of the equation. Then, plug in these roots into the expression $\dfrac{|b|+|c|}{a}$ and simplify to find the minimum value.

## 3. Can the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation be negative?

Yes, the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation can be negative. This means that the curve of the cubic equation intersects the x-axis at a point that is below the x-axis.

## 4. Is the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation always unique?

No, the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation is not always unique. If the cubic equation has multiple real roots, then there can be multiple values of $\dfrac{|b|+|c|}{a}$ that result in the minimum distance between the curve and the x-axis.

## 5. Can the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation be found without using the cubic formula?

Yes, the minimum value of $\dfrac{|b|+|c|}{a}$ from the roots of a cubic equation can also be found using calculus. By taking the derivative of the cubic equation and setting it equal to 0, you can find the critical points which will give you the minimum value of $\dfrac{|b|+|c|}{a}$.

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