Solve lambda^2(x-4)-4lambda=4(x-lambda^2)

$$\lambda ^ { 2 } ( x - 4 ) - 4 \lambda = 4 ( x - \lambda ^ { 2 } )$$

Solve for x (complex solution)

$$x=\frac{4\lambda }{\lambda ^{2}-4}$$

$$\lambda \neq -2\text{ and }\lambda \neq 2$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda ^{2} by x-4.

$$\lambda ^{2}x-4\lambda ^{2}-4\lambda =4\left(x-\lambda ^{2}\right)$$

Use the distributive property to multiply 4 by x-\lambda ^{2}.

$$\lambda ^{2}x-4\lambda ^{2}-4\lambda =4x-4\lambda ^{2}$$

Subtract 4x from both sides.

$$\lambda ^{2}x-4\lambda ^{2}-4\lambda -4x=-4\lambda ^{2}$$

Add 4\lambda ^{2} to both sides.

$$\lambda ^{2}x-4\lambda -4x=-4\lambda ^{2}+4\lambda ^{2}$$

Combine -4\lambda ^{2} and 4\lambda ^{2} to get 0.

$$\lambda ^{2}x-4\lambda -4x=0$$

Add 4\lambda to both sides. Anything plus zero gives itself.

$$\lambda ^{2}x-4x=4\lambda$$

Combine all terms containing x.

$$\left(\lambda ^{2}-4\right)x=4\lambda$$

Divide both sides by \lambda ^{2}-4.

$$\frac{\left(\lambda ^{2}-4\right)x}{\lambda ^{2}-4}=\frac{4\lambda }{\lambda ^{2}-4}$$

Dividing by \lambda ^{2}-4 undoes the multiplication by \lambda ^{2}-4.

$$x=\frac{4\lambda }{\lambda ^{2}-4}$$

Solve for x

$$x=\frac{4\lambda }{\lambda ^{2}-4}$$

$$|\lambda |\neq 2$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda ^{2} by x-4.

$$\lambda ^{2}x-4\lambda ^{2}-4\lambda =4\left(x-\lambda ^{2}\right)$$

Use the distributive property to multiply 4 by x-\lambda ^{2}.

$$\lambda ^{2}x-4\lambda ^{2}-4\lambda =4x-4\lambda ^{2}$$

Subtract 4x from both sides.

$$\lambda ^{2}x-4\lambda ^{2}-4\lambda -4x=-4\lambda ^{2}$$

Add 4\lambda ^{2} to both sides.

$$\lambda ^{2}x-4\lambda -4x=-4\lambda ^{2}+4\lambda ^{2}$$

Combine -4\lambda ^{2} and 4\lambda ^{2} to get 0.

$$\lambda ^{2}x-4\lambda -4x=0$$

Add 4\lambda to both sides. Anything plus zero gives itself.

$$\lambda ^{2}x-4x=4\lambda$$

Combine all terms containing x.

$$\left(\lambda ^{2}-4\right)x=4\lambda$$

Divide both sides by \lambda ^{2}-4.

$$\frac{\left(\lambda ^{2}-4\right)x}{\lambda ^{2}-4}=\frac{4\lambda }{\lambda ^{2}-4}$$

Dividing by \lambda ^{2}-4 undoes the multiplication by \lambda ^{2}-4.

$$x=\frac{4\lambda }{\lambda ^{2}-4}$$

Solve for λ

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