$a=\sqrt{\sqrt{b^{4}+b^{2}+c^{9}+c^{4}+4\left(bc\right)^{2}}+b^{2}+c^{2}}$
$a=-\sqrt{\sqrt{b^{4}+b^{2}+c^{9}+c^{4}+4\left(bc\right)^{2}}+b^{2}+c^{2}}$
$a=-\sqrt{-\sqrt{b^{4}+b^{2}+c^{9}+c^{4}+4\left(bc\right)^{2}}+b^{2}+c^{2}}$
$a=\sqrt{-\sqrt{b^{4}+b^{2}+c^{9}+c^{4}+4\left(bc\right)^{2}}+b^{2}+c^{2}}$

$b=\sqrt{\frac{a^{4}-c^{9}-2\left(ac\right)^{2}}{2a^{2}+2c^{2}+1}}$
$b=-\sqrt{\frac{a^{4}-c^{9}-2\left(ac\right)^{2}}{2a^{2}+2c^{2}+1}}\text{, }c<-1\text{ or }\left(c\leq 0\text{ and }|a|\leq \sqrt{-\sqrt{c^{9}+c^{4}}+c^{2}}\right)\text{ or }|a|\geq \sqrt{\sqrt{c^{9}+c^{4}}+c^{2}}$