How to find the roots for x^3+2x+2

blogmarxisteFAu

blogmarxisteFAu

Answered question

2022-12-04

How to find the roots for x 3 + 2 x + 2

Answer & Explanation

Henry Arellano

Henry Arellano

Beginner2022-12-05Added 12 answers

Let x = u + v
Hence,
u 3 + v 3 + 3 u v ( u + v ) + 2 ( u + v ) + 2 = 0.
Now, let u 3 + v 3 + 2 = 0
Thus, since u + v 0, we obtain
3 u v + 2 = 0
or
u 3 v 3 = 8 27 ,
which says u 3 and v 3 they are roots of the following equation
t 2 + 2 t 8 27 = 0.
t 1 = 1 + 35 27 , t 2 = 1 35 27 , which gives an unique real root
1 + 35 27 3 + 1 35 27 3 .
By the Viete's theorem we can get two complex roots:
x 2 = u + v 2 + ( u + v ) 3 2 i
and
x 3 = u + v 2 ( u + v ) 3 2 i .

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