how to reduce and equation to a quadratic equation?

Question

Hi everyone, I was studying the cyclotomic equation and I came across an example:
 
x^4 + x^3 + x^2 + x +1 = 0 ; this equation cannot be factorised neither using Ruffini's rule nor replacing an x with t. The book said that it is sufficient to divide all by x^2 and rearrange the terms to obtain --> x^2 + 1/x^2 + x + 1/x + 1 = 0, then let w be equal to x + 1/x, and the equation (I also have verified) becomes w^2 + w -1 = 0.
 
My question is: does it exist a formula (or a procedure) that allows to reduce an equation with a degree greater than 2 to a quadratic? Procedures other than replace x^4 with t and x^2 with t.
 In fact there would be Abel-Ruffini theorem to take into account...which says that there is not a general algebraic solution for quintic equations and higher degrees...
 
Could you help me please?

javi · Accepted Answer

as far as I know, no, it doesn't exist such thing.

Anonymous · Answer

I am definitely no mathematician and I not really sure what you want but I came up with was

x^4 + x^3 + x^2 + x + 1 = 0

= x^4 + x^3 + x^2 + x  = -1

= (x^4 + x^3 + x^2 + x)  = -1  (take out x)          

= x(x^3 + x^2 + x + 1 ) = -1     (divide by x)

= x^3 + x^2 + x + 1 = -1/x

= x^3 + x^2 + x = -1/x – 1

I am not sure if it makes any sense but that was the lowest I could think of

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how to reduce and equation to a quadratic equation?

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