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 Gustav December 5th, 2010 05:09 AM

Complex equation

I need a help with this.

$p,q \in C,\ q \neq 0$
Prove that if complex roots $x_1$ and $x_2$ of equation $x^2 + px + q^2= 0$ have their absolute values equal ( $|x_1|=|x_2|\$ then $\frac{p}{q}$ is a real number.

 DLowry December 5th, 2010 07:41 AM

Re: Complex equation

 skipjack December 5th, 2010 08:07 AM

So?

 DLowry December 5th, 2010 12:44 PM

Re:

Quote:
 Originally Posted by skipjack So?
Are you asking me or him?

 skipjack December 5th, 2010 09:23 PM

You. There is a reasonably short proof using trigonometry, but I don't see any simple method based on the quadratic formula.

 Gustav December 5th, 2010 10:11 PM

Re: Complex equation

And could you please show me both way to do it? Especially trigonometry one.

Common quadratic formula works, but from the result I am unable to work out the proof.

 skipjack December 5th, 2010 11:19 PM

Let $x_{\small1}\,=\,r\,\text{cis}(a),\ x_{\small2}\,=\,r\,\text{cis}(b),$ where "cis" means "cos + i sin" and $a$ and $b$ are real, then
p $=\,-r(\text{cis}(a)\,+\,\text{cis}(b))$ and q² $=\,r^{\small2}\text{cis}(a)\text{cis}(b)\,=\,r^{\s mall2}\text{cis}(a+b).$
Now it's fairly easy to show that p²/q² > 0.

 Gustav December 6th, 2010 01:29 AM

Re: Complex equation

But I have to prove that p/q is real number, not that p*p/q*q > 0

Or did I understand badly to it?

 DLowry December 6th, 2010 05:35 AM

Re:

Quote:
 Originally Posted by skipjack Let $x_{\small1}\,=\,r\,\text{cis}(a),\ x_{\small2}\,=\,r\,\text{cis}(b),$ where "cis" means "cos + i sin" and $a$ and $b$ are real, then p $=\,-r(\text{cis}(a)\,+\,\text{cis}(b))$ and q² $=\,r^{\small2}\text{cis}(a)\text{cis}(b)\,=\,r^{\s mall2}\text{cis}(a+b).$ Now it's fairly easy to show that p²/q² > 0.
Ah, this is much shorter than the way I had in mind. I yield for your victory.

To Gustav, consider what we now have. If p^2/q^2 = a > 0, then we also have that p^2/q^2 - a = 0. (I chose this form because I think it's easiest to see).

 Gustav December 6th, 2010 08:31 AM

Re: Complex equation

But how can we say that a and b are real?

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