Solve f^2(x)+f(x)+1=0

$$f ^ { 2 } ( x ) + f ( x ) + 1 = 0$$

Solve for x

$$x=-\frac{1}{f\left(f+1\right)}$$

$$f\neq -1\text{ and }f\neq 0$$

Steps for Solving Linear Equation

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$f^{2}x+fx=-1$$

Combine all terms containing x.

$$\left(f^{2}+f\right)x=-1$$

Divide both sides by f^{2}+f.

$$\frac{\left(f^{2}+f\right)x}{f^{2}+f}=-\frac{1}{f^{2}+f}$$

Dividing by f^{2}+f undoes the multiplication by f^{2}+f.

$$x=-\frac{1}{f^{2}+f}$$

Divide -1 by f^{2}+f.

$$x=-\frac{1}{f\left(f+1\right)}$$

Solve for f (complex solution)

$$f=\frac{\sqrt{x^{2}-4x}}{2x}-\frac{1}{2}$$

$$f=-\frac{\sqrt{x^{2}-4x}}{2x}-\frac{1}{2}\text{, }x\neq 0$$

Solve for f

$$f=\frac{\sqrt{x^{2}-4x}}{2x}-\frac{1}{2}$$

$$f=-\frac{\sqrt{x^{2}-4x}}{2x}-\frac{1}{2}\text{, }x<0\text{ or }x\geq 4$$