$\left\{\begin{matrix}a=-\frac{b^{2}}{x}\text{, }&b\neq 0\text{ and }x\neq -b\text{ and }x\neq 0\text{ and }x\neq b\\a\neq 0\text{, }&b=0\text{ and }x=0\end{matrix}\right.$

$b=-i\sqrt{a}\sqrt{x}$
$b=i\sqrt{a}\sqrt{x}\text{, }\left(arg(-i\sqrt{a})\geq \pi \text{ and }a\neq 0\text{ and }arg(i\sqrt{a})\geq \pi \right)\text{ or }\left(x\neq -a\text{ and }a\neq 0\right)$

$\left\{\begin{matrix}a=-\frac{b^{2}}{x}\text{, }&b\neq 0\text{ and }x\neq 0\text{ and }|x|\neq |b|\\a\neq 0\text{, }&b=0\text{ and }x=0\end{matrix}\right.$

$b=\sqrt{-ax}$
$b=-\sqrt{-ax}\text{, }\left(a>0\text{ or }x\geq 0\right)\text{ and }a\neq 0\text{ and }\left(x\leq 0\text{ or }a<0\right)\text{ and }x\neq -a$