Substituting p=x^2 for Solving Polynomial Equation by Hand

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Homework Help Overview

The discussion revolves around solving the polynomial equation (1/2)x^4 - x^2 - 1 = 0, specifically through the substitution of p = x^2 to simplify the problem.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the substitution method to transform the original polynomial into a quadratic equation in terms of p. There is an examination of the implications of this substitution and the resulting expressions.

Discussion Status

The discussion includes confirmation of the substitution approach and the transformation into a quadratic form. Some participants have provided expressions derived from this method, indicating a progression in the exploration of solutions.

Contextual Notes

There may be constraints related to solving the equation by hand, as implied by the original poster's inquiry. The discussion does not clarify all assumptions or the completeness of the proposed methods.

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is there a way to solve this by hand
(1/2)x^4-x^2-1=0
 
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Yes substitute p=x^2.
 


(1/2)x^4-x^2-1=(1/2)p^2-p-1=0

quadratic in p

[1+||- (1+2)^(1/2) ] = 1+||-3^(1/2) = p = x^2

x=+||-[1+||-3^(1/2) ]^(1/2) =+||-[1+3^(1/2) ]^(1/2)
 


That's correct.
 

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