Solve -f(x)=x/x^2-3x+2

$$- f ( x ) = \frac { x } { x ^ { 2 } - 3 x + 2 }$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=\frac{1}{\left(1-x\right)\left(x-2\right)}\text{, }&x\neq 1\text{ and }x\neq 2\\f\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(x-1\right).

$$\left(-f\right)x\left(x-2\right)\left(x-1\right)=x$$

Use the distributive property to multiply \left(-f\right)x by x-2.

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

Multiply -2 and -1 to get 2.

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

Use the distributive property to multiply \left(-f\right)x^{2}+2fx by x-1.

$$\left(-f\right)x^{3}-\left(-f\right)x^{2}+2fx^{2}-2fx=x$$

Multiply -1 and -1 to get 1.

$$\left(-f\right)x^{3}+fx^{2}+2fx^{2}-2fx=x$$

Combine fx^{2} and 2fx^{2} to get 3fx^{2}.

$$\left(-f\right)x^{3}+3fx^{2}-2fx=x$$

Reorder the terms.

$$-fx^{3}+3fx^{2}-2fx=x$$

Combine all terms containing f.

$$\left(-x^{3}+3x^{2}-2x\right)f=x$$

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

$$\frac{\left(-x^{3}+3x^{2}-2x\right)f}{-x^{3}+3x^{2}-2x}=\frac{x}{-x^{3}+3x^{2}-2x}$$

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

$$f=\frac{x}{-x^{3}+3x^{2}-2x}$$

Divide x by -x^{3}+3x^{2}-2x.

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

Solve for f

$$\left\{\begin{matrix}f=\frac{1}{\left(1-x\right)\left(x-2\right)}\text{, }&x\neq 2\text{ and }x\neq 1\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(x-1\right).

$$\left(-f\right)x\left(x-2\right)\left(x-1\right)=x$$

Use the distributive property to multiply \left(-f\right)x by x-2.

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

Multiply -2 and -1 to get 2.

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

Use the distributive property to multiply \left(-f\right)x^{2}+2fx by x-1.

$$\left(-f\right)x^{3}-\left(-f\right)x^{2}+2fx^{2}-2fx=x$$

Multiply -1 and -1 to get 1.

$$\left(-f\right)x^{3}+fx^{2}+2fx^{2}-2fx=x$$

Combine fx^{2} and 2fx^{2} to get 3fx^{2}.

$$\left(-f\right)x^{3}+3fx^{2}-2fx=x$$

Reorder the terms.

$$-fx^{3}+3fx^{2}-2fx=x$$

Combine all terms containing f.

$$\left(-x^{3}+3x^{2}-2x\right)f=x$$

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

$$\frac{\left(-x^{3}+3x^{2}-2x\right)f}{-x^{3}+3x^{2}-2x}=\frac{x}{-x^{3}+3x^{2}-2x}$$

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

$$f=\frac{x}{-x^{3}+3x^{2}-2x}$$

Divide x by -x^{3}+3x^{2}-2x.

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

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=-\frac{\sqrt{f\left(f-4\right)}-3f}{2f}\text{; }x=\frac{\sqrt{f\left(f-4\right)}+3f}{2f}\text{, }&f\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=-\frac{\sqrt{f\left(f-4\right)}-3f}{2f}\text{; }x=\frac{\sqrt{f\left(f-4\right)}+3f}{2f}\text{, }&f<0\text{ or }f\geq 4\end{matrix}\right.$$