How does one solve this equation. I would like to see the solution of this problem in steps.
$z\cdot\bar{z}=\left|3\cdot z \right|$
EDIT: Is it possible to solve this by converting to the form $z=a+b\cdot i$
What about the solution of this equation.
$z\cdot\bar{z}-z^{2}=1-i$
EDIT2:
$a^2+b^2-(a+b\cdot i)(a+b\cdot i) = 1 – i$
$a^2+b^2-a^2-ab\cdot i – ab\cdot i + b^2=1-i$
$2b^2-2ab\cdot i = 1-i$
And we keep in mind that two imaginary numbers are equal if their real and imaginary parts are the same.
$2b^2 = 1$ and $-2ab=-1$
So $b = \pm \frac{1}{\sqrt{2}}$
and $a=\frac{1}{2b}\Rightarrow a=\pm \frac{\sqrt{2}}{2}$.
Is this correct?
Best Answer
In steps:
$z\cdot \bar z = |z|^2$;
$|3\cdot z| = 3|z|$;
$|z|^2 = 3|z|\Leftrightarrow |z| = 0\text{ or }|z| = 3\Leftrightarrow z = 0\text{ or }z = 3\mathrm e^{i\phi}$ for $\phi\in\mathbb R$.