COMPLEX SOLUTIONS help!

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Main Question or Discussion Point

First problem
z^2+2z+1=0
where the z in 2z is the conjugate (has a little line ontop)
I just ignored the conjugate because I wasn't sure how to solve it, and I got -1 which is one of the solutions but there's also 1+2i and 1-2i which I understand because they're both conjugate of each other but I don't understand how they got it.
Second problem
z^3-3z^2+4z-12=0 given 2i is a solution... I don't even understand what they mean.
Please help!
 

Answers and Replies

7
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First problem
z^2+2z+1=0
where the z in 2z is the conjugate (has a little line ontop)
I just ignored the conjugate because I wasn't sure how to solve it, and I got -1 which is one of the solutions but there's also 1+2i and 1-2i which I understand because they're both conjugate of each other but I don't understand how they got it.
Second problem
z^3-3z^2+4z-12=0 given 2i is a solution... I don't even understand what they mean.
Please help!
As I see your problem is
[tex]z^{2} + 2 z^{*} + 1 = 0[/tex]

if you will search solution in following form [tex]z = a + i b[/tex], [tex]a, b[/tex] are both real numbers and insert it to your main equation then you will have system of two simple algebraic equations under [tex]a, b[/tex] and you'll find [tex]a = 1, b = \pm 2[/tex]. Solve your second problem in the same way and get the answer.
 
14
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I get that z=a+ib but how did you get values for a and b? I feel really stupid asking this but I dont see it. I tried solving it and then making b=0 and a=0 and I'm not getting 1 and 2 as values...
 
7
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I get that z=a+ib but how did you get values for a and b? I feel really stupid asking this but I dont see it. I tried solving it and then making b=0 and a=0 and I'm not getting 1 and 2 as values...
What are your equations for a and b?
 
HallsofIvy
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Do the algebra. If z= a+ ib, then [itex]\overline{z}[/itex]= a- ib so [itex]z^2+ 2\overline{z}+ 1= (a+ ib)^2+ 2(a- ib)+ 1= 0[/itex]. Separate the real and imaginary parts and you have two equations for a and b.
 
14
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Yeah I got that far but i'm not sure how to seperate real and imaginary parts. Is it literally just placing all the real parts and making them equal to 0 and all the imaginary parts and make them equal to 0? Sorry our lecturer didn't go through this and so I'm just left lost. :shy:
Thank you
 
HallsofIvy
Science Advisor
Homework Helper
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Yes, it literally is! If a+ bi= c+ di, then a= c and b= d. That's part of the definition of "complex number".
 
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