Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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
    
@HagenvonEitzen: How did you obtain this inequalities? Here I have a similler question. Can you help me to solve it? Thank you. math.stackexchange.com/questions/985518/… –  Nilan Oct 22 at 7:44
    
@Nilan $-a$ is the sum of the two roots, hence $\Re a>0$ is necessary. –  Hagen von Eitzen Oct 22 at 10:30
    
@HagenvonEitzen: I got that. How did you get the sufficient condition? –  Nilan Oct 23 at 7:46

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.