Finding roots and complex roots of a determinant

AI Thread Summary
The discussion focuses on finding the values of Ω from the equation involving determinants and complex roots. The initial equation simplifies to a fourth-degree polynomial, which the participants analyze for roots. Real roots are identified as ±√(2k/3m), but the challenge lies in determining the complex roots. The suggestion to use the identity a² - b² = (a + b)(a - b) is highlighted as a potential method for finding these roots. Ultimately, the complex roots are successfully identified using the notation provided by a participant.
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Homework Statement
Finding roots and complex roots of a determinant
Relevant Equations
##(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m})(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m}) - (-i\gamma\Omega)(-i\gamma\Omega) = 0##
I need to find the values of ##\Omega## where ##(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m})(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m}) - (-i\gamma\Omega)(-i\gamma\Omega) = 0##

I get ##\Omega^4 -2i\gamma \Omega^3 - \frac{4k}{3m}\Omega^2 + i\frac{4k}{3m}\gamma\Omega + \frac{4k^2}{9m^2} = 0##

I don't think this is the correct way.
I don't find a way to resolve a quadratic.
I have to factor some terms, but I don't see where.
 
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You have ##f(\Omega)^2=g(\Omega)^2##. Why don't you check all possibilities from here?
 
a^2 - b^2 = (a +b)(a - b) also holds for complex a and b...
 
I found the the reals roots which are ##\Omega = + -\sqrt{\frac{2k}{3m}}## using ##(-\Omega^2 +i\gamma\Omega + \frac{2k}{3m}))^2 = i^2\gamma^2\Omega^2##

However, I don't see how to get the complex roots.
To find the complex roots, should I replace ##i^2## by -1 ?
 
You have ##f(\Omega) =\pm g(\Omega)## or ##a=\pm b## with @pasmith 's notation.​

Now ##f(\Omega)## is quadratic, that gives you ##4## roots, two real and two complex.
 
Alright, Thanks!
I found it using Pasmith's notation.
Thanks guys!
 
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