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Easiest way to proceed is the square both sides and see what you get from there

so ... waiting for you to take a step or engage in the conversation

that gives you some of the solutions. You also need to consider the case
z + 9/z = -6

Note: Z is a Complex Number

Or is z here a complex number?

Z is a Complex Number

If z is complex, we're going to have use a different approach.

right. So in that case, what have you tried?

No Idea..

(x^2 -y^2 + 9)^2 + (2xy)^2 = 36(x^2 + y^2)

Yes, now see if you can reduce this somehow

|dw:1360344301541:dw|

Nope..i havent Studied that

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|dw:1360344710864:dw|
damn this is getting complicated.

|dw:1360344933595:dw|

very nice, yes.