Find all complex solutions of the system

(a + ic)^3 + (ia + b)^3 + (−b + ic)^3 = −6 ,

(a + ic)^2 + (ia + b)^2 + (−b + ic)^2 = 6 ,

(1 + i)a + 2ic = 0 .

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- Jul 20th 2008, 11:01 PMperashsolutions of the system
Find all complex solutions of the system

(a + ic)^3 + (ia + b)^3 + (−b + ic)^3 = −6 ,

(a + ic)^2 + (ia + b)^2 + (−b + ic)^2 = 6 ,

(1 + i)a + 2ic = 0 . - Jul 21st 2008, 01:58 AMMoo
Hello,

Let's study (3).

Note that .

Therefore :

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Substituting in (1) and (2), we get the following system :

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Now, develop (2').

Note that

Divide by 2 :

**Equating real and imaginary parts :**<< here is the mistake... because b and c are complex numbers, it doesn't work. Pooh

From the second equation, we get

2 possibilities :

- b is a real number

- b is a complex number

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**b is a real number**

Therefore, the equation (4) is impossible and the**possible**solutions are :

Substituting in (1'), one can see that this yields .

So b can't be a real number.

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**b is a complex number**

So :

[tex]b'=i \sqrt{3}[/tex]

:

If [tex]b'=i \sqrt{3}[/tex], and

If [tex]b''=-i \sqrt{3}[/tex], and

The couples of solutions are

We'll substitute in the equation (1') :

~~~~~~~~~~~~~~~~~~~~

First recall or note down a few useful things :

(example)

Let's take the b',c' solutions.

We want this to be equal to -6.

So the solution a',b',c' doesn't work.

The same goes for a'',b'',c''.

So there's no solution... (Crying)(Crying)(Crying)(Crying)

I don't know if I'm wrong or if there is really no solution, but all this work for.. nothing, that's sad(istic) :(

Edit : Ok, calculator gives a=1+i, b=2-i, c=-1 and a=1+i, b=-1-i, c=-1 and a=-2-2i, b=-1+2i, c=2

I'll have a look later.. - Jul 21st 2008, 03:58 AMflyingsquirrel
Hi

I deleted my previous post because I found a mistake in my work but fixing it didn't change the method I used to solve the problem so here it is :

Letting , and the system becomes :

Once you've solve this for and you'll get the values of and by solving

In case you wonder, I'm not sure this method will give you the result more quickly than Moo's one. Good luck ! - Jul 21st 2008, 06:25 AMNonCommAlg