Conjugate quadratic equations

Question

If a quadratic equation is defined such that ,

$Z^2 + aZ + b =0$ where $a,b$ belong to the set of complex numbers ,

why is this the same thing as $\overline{Z^2 + aZ + b}=0$ ?

Why do both the equations mean the same thing ? Which formula is used here? Please help.

Answer 1

Because $\overline{0}=0$.

Also, $$\overline{Z^2+aZ+b}=\overline{Z}^2+\overline{a}\overline{Z}+\overline{b}$$ if you wish.

For $\{x,y\}\subset\mathbb R$ we have $$\overline{x+yi}=x-yi$$ of course.

Answer 2

Simply because for any complex number $\;z=0\iff \bar z=0\iff|z|=0$.

Answer 3

They mean the same thing because if two complex numbers (in this case $Z^2 + aZ + b$ and $0$) are the same, then their conjugates are the same as well, and vice versa. So "take the conjugate on both sides" is an entirely valid thing to do to an equation, similar to "multiply both sides by $2$" and "add $4$ to both sides".