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# Halp!

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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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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