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Consider $x^2+ax+b=0$, where $x$ is the variable and $a,b$ are complex coefficients. Is there any condition on $a$ and $b$ which makes sure the roots of the equation have negative real parts?

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Hanna, you have zero (!) "accept rate", which means that you haven't accepted as "best" any of the answers given to any of your questions. This seems to imply you don't like the answers you receive here and many people will be discouraged to try to help you. For any questions about this "accept rate" you can read here: meta.math.stackexchange.com/questions/tagged/accepted-answer –  DonAntonio Jan 10 '13 at 13:46
A necessary condition is $\Re a>0$, a sufficient condition is $4|b|<(\Re a)^2$. –  Hagen von Eitzen Jan 10 '13 at 13:48
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1 Answer

Let $\,z_1:=a+bi\,,\,z_2:=c+di\in\Bbb C\,$ be the two roots, then using Viete's formulae:



Now compare real and imaginary parts in the two expressions above.

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