Solve (lambda-1)x^2+lambda^2x-lambda=0

$$( \lambda - 1 ) x ^ { 2 } + \lambda ^ { 2 } x - \lambda = 0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{\lambda \left(\lambda ^{3}+4\lambda -4\right)}-\lambda ^{2}}{2\left(\lambda -1\right)}\text{; }x=-\frac{\sqrt{\lambda \left(\lambda ^{3}+4\lambda -4\right)}+\lambda ^{2}}{2\left(\lambda -1\right)}\text{, }&\lambda \neq 1\\x=1\text{, }&\lambda =1\end{matrix}\right.$$

Solve for λ (complex solution)

$$\left\{\begin{matrix}\lambda =\frac{\sqrt{x^{4}+4x^{3}-2x^{2}+1}-x^{2}+1}{2x}\text{; }\lambda =\frac{-\sqrt{x^{4}+4x^{3}-2x^{2}+1}-x^{2}+1}{2x}\text{, }&x\neq 0\\\lambda =0\text{, }&x=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{\lambda \left(\lambda ^{3}+4\lambda -4\right)}-\lambda ^{2}}{2\left(\lambda -1\right)}\text{; }x=-\frac{\sqrt{\lambda \left(\lambda ^{3}+4\lambda -4\right)}+\lambda ^{2}}{2\left(\lambda -1\right)}\text{, }&\lambda \neq 1\text{ and }\lambda ^{4}+4\lambda ^{2}-4\lambda \geq 0\\x=1\text{, }&\lambda =1\end{matrix}\right.$$

Solve for λ

$$\left\{\begin{matrix}\lambda =\frac{\sqrt{x^{4}+4x^{3}-2x^{2}+1}-x^{2}+1}{2x}\text{; }\lambda =\frac{-\sqrt{x^{4}+4x^{3}-2x^{2}+1}-x^{2}+1}{2x}\text{, }&x\neq 0\text{ and }x^{4}+4x^{3}-2x^{2}+1\geq 0\\\lambda =0\text{, }&x=0\end{matrix}\right.$$