Επίλυση abcx^2-left(a^2b^2-c^2right)x+abc=0

$$abc { x }^{ 2 } - \left( { a }^{ 2 } { b }^{ 2 } - { c }^{ 2 } \right) x+abc=0$$

Λύση ως προς a (complex solution)

$$\left\{\begin{matrix}a\in \mathrm{C}\text{, }&\left(b=0\text{ or }x=0\right)\text{ and }\left(b=0\text{ or }c=0\right)\text{ and }\left(x=0\text{ or }c=0\right)\\a=\frac{\sqrt{c^{2}\left(x^{4}+6x^{2}+1\right)}+cx^{2}+c}{2bx}\text{; }a=-\frac{\sqrt{c^{2}\left(x^{4}+6x^{2}+1\right)}-cx^{2}-c}{2bx}\text{, }&b\neq 0\text{ and }x\neq 0\\a=0\text{, }&x=0\text{ and }c\neq 0\end{matrix}\right,$$

Λύση ως προς b (complex solution)

$$\left\{\begin{matrix}b\in \mathrm{C}\text{, }&\left(a=0\text{ or }x=0\right)\text{ and }\left(a=0\text{ or }c=0\right)\text{ and }\left(x=0\text{ or }c=0\right)\\b=\frac{\sqrt{c^{2}\left(x^{4}+6x^{2}+1\right)}+cx^{2}+c}{2ax}\text{; }b=-\frac{\sqrt{c^{2}\left(x^{4}+6x^{2}+1\right)}-cx^{2}-c}{2ax}\text{, }&a\neq 0\text{ and }x\neq 0\\b=0\text{, }&x=0\text{ and }c\neq 0\end{matrix}\right,$$

Λύση ως προς a

$$\left\{\begin{matrix}a\in \mathrm{R}\text{, }&\left(b=0\text{ or }x=0\right)\text{ and }\left(b=0\text{ or }c=0\right)\text{ and }\left(x=0\text{ or }c=0\right)\\a=\frac{c\left(\sqrt{x^{4}+6x^{2}+1}+x^{2}+1\right)}{2bx}\text{; }a=-\frac{c\left(\sqrt{x^{4}+6x^{2}+1}-x^{2}-1\right)}{2bx}\text{, }&b\neq 0\text{ and }x\neq 0\\a=0\text{, }&x=0\text{ and }c\neq 0\end{matrix}\right,$$

Λύση ως προς b

$$\left\{\begin{matrix}b\in \mathrm{R}\text{, }&\left(a=0\text{ or }x=0\right)\text{ and }\left(a=0\text{ or }c=0\right)\text{ and }\left(x=0\text{ or }c=0\right)\\b=\frac{c\left(\sqrt{x^{4}+6x^{2}+1}+x^{2}+1\right)}{2ax}\text{; }b=-\frac{c\left(\sqrt{x^{4}+6x^{2}+1}-x^{2}-1\right)}{2ax}\text{, }&a\neq 0\text{ and }x\neq 0\\b=0\text{, }&x=0\text{ and }c\neq 0\end{matrix}\right,$$