Find all solution in <mi mathvariant="double-struck">R for the following system of equations:

skottyrottenmf 2022-05-26 Answered
Find all solution in R for the following system of equations:
{ x + 3 x y x 2 + y 2 = 3 y x + 3 y x 2 + y 2 = 0
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Answers (2)

tatoas9f
Answered 2022-05-27 Author has 11 answers
Complexify. Let z = x + i y. Then for x 2 + y 2 0:
( x + 3 x y x 2 + y 2 ) + i ( y x + 3 y x 2 + y 2 ) = ( 3 + i 0 ) ( x + i y ) + ( 3 i ) ( x i y ) x 2 + y 2 = 3 z + 3 i z = 3 z 2 3 z + ( 3 i ) = 0 ( z 3 2 ) 2 = i 3 4 z = 3 ± 4 i 3 2 .
Now, ( 1 + 2 i ) 2 = 3 + 4 i, so the solutions are
z = 3 ± ( 1 + 2 i ) 2 ,
either z = 2 + i or z = 1 i, which, in real form, yields ( x , y ) = ( 2 , 1 ) or ( x , y ) = ( 1 , 1 ).
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Aidyn Cox
Answered 2022-05-28 Author has 5 answers
Great solution and answer. I didn't know that complexifying will provide a real solution. Thanks again.
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