Solving complex quartic equation

dourtuntellorvl

dourtuntellorvl

Answered question

2022-06-15

Solving complex quartic equation z 5 = 1

Answer & Explanation

Donavan Mack

Donavan Mack

Beginner2022-06-16Added 24 answers

Factorize the equation as follows
z 5 1 = ( z 1 ) ( z 4 + z 3 + z 2 + z + 1 ) = ( z 1 ) z 2 ( z 2 + z + 1 + 1 z + 1 z 2 ) = ( z 1 ) z 2 ( ( z + 1 z ) 2 + ( z + 1 z ) 1 ) = ( z 1 ) z 2 ( z + 1 z 1 + 5 2 ) ( z + 1 z 1 5 2 ) = ( z 1 ) ( z 2 1 + 5 2 z + 1 ) ( z 2 1 5 2 z + 1 )
which yields the solutions
x = 1 , 1 4 ( 1 + 5 ± i 10 2 5 ) , 1 4 ( 1 5 ± i 10 + 2 5 )

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