Understanding the Quadratic Formula: Why the Negative Root is Not Considered

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Homework Help Overview

The discussion revolves around the cubic equation x^3 - 4x^2 + 4x + 2 = 0 and the application of the quadratic formula to find its roots. Participants are particularly focused on the treatment of negative roots in the calculations.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster questions why only the positive root is considered in the quadratic formula application and why another negative root is not used in the calculations. Other participants suggest that both roots can be used and inquire if the original poster has tried this approach.

Discussion Status

Participants are actively engaging with the original poster's questions, exploring the implications of using different roots. There is a back-and-forth regarding the calculations and potential outcomes, indicating a productive exploration of the topic without reaching a consensus.

Contextual Notes

There is an emphasis on the nature of the roots being negative and the implications this has for the calculations being discussed. The original poster's inquiry suggests a need for clarification on the assumptions made in the problem setup.

davon806
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Homework Statement


Please see the method of solving the equation x^3 -4x^2 + 4x +2 = 0 from the attached.
In the red brackets,can anyone explain why the negative root is not involved in the calculation?
by the quadratic formula,
z = [y±√(y^2 - 4k)]/2
so why only y+√(y^2 - 4k) is considered?
and in the second bracket,again,why the another root of u ,(-1/27)(-35 - 3√129),is not used?
Both roots are negative.
Thx.

Homework Equations





The Attempt at a Solution

 

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hi davon806! :wink:
davon806 said:
In the red brackets,can anyone explain why the negative root is not involved in the calculation?

and in the second bracket,again,why the another root of u ,(-1/27)(-35 - 3√129),is not used?

you can use the "negative" root in either or both cases, and it should give the same result in the end …

have you tried it? :smile:
 
As there are 2 solutions of z and 2 solutions of u,
there are 4 possible answers for x
z = [y ± √(y^2 - 4k)]/2 , u = 1/27(-35 ± 3√129)
if both of the signs are +,or both signs are -,the value of x would be:
x = 4/3 - 4/[3(35-3√129)^(1/3) - 1/3(35-3√129)^(1/3)]
if z = [y+√(y^2 - 4k)]/2 and u = (-35-3√129)/27 ,
x = 4/3 - 4/[3(35+3√129)^(1/3) - 1/3(35+3√129)^(1/3)]

What's wrong?Thx.
 
hi davon806! :smile:
davon806 said:
x = 4/3 - 4/[3(35-3√129)^(1/3) - 1/3(35-3√129)^(1/3)]
if z = [y+√(y^2 - 4k)]/2 and u = (-35-3√129)/27 ,
x = 4/3 - 4/[3(35+3√129)^(1/3) - 1/3(35+3√129)^(1/3)]

ah, but 1/(35-3√129) = (35+3√129)/64 :wink:
 

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