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Let $\omega$ be a complex number such that $|\omega| = 1,$ and the equation \[z^2 + z + \omega = 0\]has a pure imaginary root $z.$ Find $\omega + \overline{\omega}.$

 Jan 26, 2021
 #1
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Sorry I didn't give what I think about this question, here was my go at it, 

 

First of all we make the unknowns $a, b, z$

in which $|a+bi|=\omega$,

solving for magnitude we get $a^2+b^2=1$

That is our first equation,

Our second equation is 


$z^2+z+(a+bi)=0$

this is because $\omega=a+bi$, we can plug $a+bi$ in to make our third equation,

finally we find use the pure imaginary root $z$ as the last equation

 

\(z = {-1 \pm \sqrt{1^2-4(1)4(a+bi)} \over 2}\)

This is because the root is $z$, and the 

$a=1$

$b=1$

and 

$c=(a+bi)$

from the second equation: $z^2+z+(a+bi)=0$


 

and those are three equations with 3 unkowns, did I make a mistake?
 

 Jan 26, 2021
edited by Guest  Jan 26, 2021
edited by Guest  Jan 26, 2021
 #2
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\(a^2+b^2=1\)

 

So a and b can't both equal 1.

Melody  Jan 28, 2021

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