Solve y=(a^2-1)(x^2-1)+(a-1)(x-1)

$$y = ( a ^ { 2 } - 1 ) ( x ^ { 2 } - 1 ) + ( a - 1 ) ( x - 1 )$$

Solve for a (complex solution)

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

Solve for x (complex solution)

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

Solve for a

$$\left\{\begin{matrix}a=\frac{\sqrt{\left(x-1\right)\left(4x^{3}+8x^{2}+4xy-3x+4y-9\right)}+x-1}{2\left(1-x^{2}\right)}\text{; }a=\frac{-\sqrt{\left(x-1\right)\left(4x^{3}+8x^{2}+4xy-3x+4y-9\right)}+x-1}{2\left(1-x^{2}\right)}\text{, }&\left(y=-\frac{\left(x-1\right)\left(2x+3\right)^{2}}{4\left(x+1\right)}\text{ and }x\neq -1\text{ and }|x|<1\right)\text{ or }\left(y\leq -\frac{\left(x-1\right)\left(2x+3\right)^{2}}{4\left(x+1\right)}\text{ and }x>-1\text{ and }|x|<1\right)\text{ or }\left(y\geq -\frac{\left(x-1\right)\left(2x+3\right)^{2}}{4\left(x+1\right)}\text{ and }|x|>1\right)\text{ or }\left(y=-\frac{\left(x-1\right)\left(2x+3\right)^{2}}{4\left(x+1\right)}\text{ and }|x|\neq 1\right)\\a=\frac{2-y}{2}\text{, }&x=-1\\a\in \mathrm{R}\text{, }&x=1\text{ and }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(a-1\right)\left(4ay+4y+4a^{3}+8a^{2}-3a-9\right)}+a-1}{2\left(1-a^{2}\right)}\text{; }x=\frac{-\sqrt{\left(a-1\right)\left(4ay+4y+4a^{3}+8a^{2}-3a-9\right)}+a-1}{2\left(1-a^{2}\right)}\text{, }&\left(y=-\frac{\left(a-1\right)\left(2a+3\right)^{2}}{4\left(a+1\right)}\text{ and }a\neq -1\text{ and }|a|<1\right)\text{ or }\left(y\leq -\frac{\left(a-1\right)\left(2a+3\right)^{2}}{4\left(a+1\right)}\text{ and }a>-1\text{ and }|a|<1\right)\text{ or }\left(y\geq -\frac{\left(a-1\right)\left(2a+3\right)^{2}}{4\left(a+1\right)}\text{ and }|a|>1\right)\text{ or }\left(y=-\frac{\left(a-1\right)\left(2a+3\right)^{2}}{4\left(a+1\right)}\text{ and }|a|\neq 1\right)\\x=\frac{2-y}{2}\text{, }&a=-1\\x\in \mathrm{R}\text{, }&a=1\text{ and }y=0\end{matrix}\right.$$