How do i further solve the following complex equation:
$$ z\cdot \bar{z} + z + \bar{z} + i\cdot z - \overline{i \cdot z} = 9 + 4i $$ $$ a^{2} - b^{2} + 2a - 2b = 9 + 4i$$
How do i solve from here on ?
How do i further solve the following complex equation:
$$ z\cdot \bar{z} + z + \bar{z} + i\cdot z - \overline{i \cdot z} = 9 + 4i $$ $$ a^{2} - b^{2} + 2a - 2b = 9 + 4i$$
How do i solve from here on ?
To solve the equation $$z\bar{z}+z+\bar{z}+iz-\overline{iz}=9+4i,$$ we can proceed much as you did. Let $z=a+ib$.
Then $z\bar{z}=(a+ib)(a-ib)=a^2+b^2$.
We have $iz=-b+ia$, so its conjugate is $-b-ia$. Our expression is therefore equal to $$a^2+b^2+(a+ib)+(a-ib)+(-b+ia)-(-b-ia),$$ which simplifies to $a^2+b^2+2a+2ia$.
This is $9+4i$ precisely if the imaginary parts match and the real parts match. We end up with the equations $2a=4$ and $a^2+b^2+2a=9$. Now $a$ and then $b$ are easy to find.
Alternative approach: the real part of both hands must be equal: $$z\bar z + z+\bar z = 9.$$ The imaginary part of both hands must be equal: $$i(z+\bar z)=4i.$$
So $z$ and $\bar z$ are two numbers; their sum is $4$ and their product is $9-4=5$. So they satisfy $$ z^2 - 4z + 5 = 0.$$ This you can solve.