1. ## solving complex numbers

This may sound like silly question, but I've made multiple attempts and can't figure out what do do with this question:

Find all solutions to z2 + 4z' + 4 = 0 where z is a complex number.
*z' being the conjugate

2. ## Re: solving complex numbers

Originally Posted by cellae
This may sound like silly question, but I've made multiple attempts and can't figure out what do do with this question:

Find all solutions to z2 + 4z' + 4 = 0 where z is a complex number.
*z' being the conjugate
Let $\displaystyle z=a+bi$ , then :

$\displaystyle (a^2-b^2)+2abi+4(a-bi)+4=0$

Hence :

$\displaystyle \begin{cases}a^2-b^2+4a+4=0 \\2ab-4b=0 \end{cases}$

3. ## Re: solving complex numbers

Why, thank you !!

4. ## Re: solving complex numbers

In this case z=z*.

5. ## Re: solving complex numbers

.. Could you extrapolate?

6. ## Re: solving complex numbers

Extrapolate?

$\displaystyle (z+2)^2=z^2+4z+4$

So with Im(z)=0, z=z' and you have the solution z=2.

7. ## Re: solving complex numbers

Originally Posted by cellae
.. Could you extrapolate?
If a polynomial has real coefficients then if $\displaystyle z_0$ is a root then so is $\displaystyle \overline{z_0}$.
Thus all that needs to be done is to solve $\displaystyle z^2+4z+4=0~.$

8. ## Re: solving complex numbers

Ok I get that.
I was confused initially because I wasn't really sure how to go about the conjagate.
That being because it's the imaginary number that get's its sign swapped. So Wasn't sure if i could change the sign of a whole complex number.

Brilliant