MHB Complex Numbers - Number of Solutions

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The equation z² + i̅z = -2 has only two imaginary solutions, which can be derived by substituting z = a + bi and separating the real and imaginary parts. This leads to a system of equations that confirms the existence of two solutions. Regarding part B, the proposed vertices Z1+3, Z2+3, Z1+i, and Z2+i do not form a rectangle but rather a parallelogram. While both shapes share properties, the distinction is important for accurate geometric representation. Understanding these concepts is crucial for solving complex number problems effectively.
Lancelot1
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Hiya all,

I need your assistance with the following problem:

A) Show that the equation

\[z^{2}+i\bar{z}=(-2)\]

has only two imaginary solutions.

B) If Z1 and Z2 are the solutions, draw a rectangle which has the following vertices:

Z1+3 , Z2+3 , Z1+i , Z2+i

I do not know how to even start. Should I try to write Z as a+bi ? Please help (Doh)
 
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Lancelot said:
A) Show that the equation
\[z^{2}+i\bar{z}=(-2)\]
has only two imaginary solutions.

B) If Z1 and Z2 are the solutions, draw a rectangle which has the following vertices:

Z1+3 , Z2+3 , Z1+i , Z2+i

I do not know how to even start. Should I try to write Z as a+bi ?
Yes, write $z = a+ib$. Then the equation becomes $(a+ib)^2 + i(a-ib) + 2 = 0.$ Now remember that if a complex number is zero then its real and imaginary parts must both be zero. That will give you two equations for the real numbers $a$ and $b$, and you should find that there are just two solutions.

I don't know what to say about part B), because as far as I can see, those four points do not form the vertices of a rectangle. (I think it should be a parallelogram.)
 
Isn't a rectangle a sort of parallelogram ? How did you see it's a parallelogram ?
 
Yes, a rectangle is a type of parallelogram. Because you said it is a rectangle, it is a parallelogram. I'm not sure that calling it a parallelogram helps though!
 
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