Solve (lambda-1)x^2+3lambdax+lambda=0

$$( \lambda - 1 ) x ^ { 2 } + 3 \lambda x + \lambda = 0$$

Solve for λ

$$\lambda =\frac{x^{2}}{x^{2}+3x+1}$$

$$x\neq \frac{\sqrt{5}-3}{2}\text{ and }x\neq \frac{-\sqrt{5}-3}{2}$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda -1 by x^{2}.

$$\lambda x^{2}-x^{2}+3\lambda x+\lambda =0$$

Add x^{2} to both sides. Anything plus zero gives itself.

$$\lambda x^{2}+3\lambda x+\lambda =x^{2}$$

Combine all terms containing \lambda .

$$\left(x^{2}+3x+1\right)\lambda =x^{2}$$

Divide both sides by x^{2}+3x+1.

$$\frac{\left(x^{2}+3x+1\right)\lambda }{x^{2}+3x+1}=\frac{x^{2}}{x^{2}+3x+1}$$

Dividing by x^{2}+3x+1 undoes the multiplication by x^{2}+3x+1.

$$\lambda =\frac{x^{2}}{x^{2}+3x+1}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{\lambda \left(5\lambda +4\right)}-3\lambda }{2\left(\lambda -1\right)}\text{; }x=-\frac{\sqrt{\lambda \left(5\lambda +4\right)}+3\lambda }{2\left(\lambda -1\right)}\text{, }&\lambda \neq 1\\x=-\frac{1}{3}\approx -0.333333333\text{, }&\lambda =1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{\lambda \left(5\lambda +4\right)}-3\lambda }{2\left(\lambda -1\right)}\text{; }x=-\frac{\sqrt{\lambda \left(5\lambda +4\right)}+3\lambda }{2\left(\lambda -1\right)}\text{, }&\left(\lambda \neq 1\text{ and }\lambda \geq 0\right)\text{ or }\lambda \leq -\frac{4}{5}\\x=-\frac{1}{3}\approx -0.333333333\text{, }&\lambda =1\end{matrix}\right.$$