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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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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:

$$ac-bd+(ad+bc)i=z_1z_2=b:=x+yi$$

$$a+c+(b+d)i=z_1+z_2=-a:=-p-qi$$

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

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