Solve xf(x)=x^2/x^2-6x+5

$$x f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 6 x + 5 }$$

Solve for f

$$\left\{\begin{matrix}f=\frac{1}{\left(x-5\right)\left(x-1\right)}\text{, }&x\neq 5\text{ and }x\neq 1\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-5\right)\left(x-1\right).

$$xfx\left(x-5\right)\left(x-1\right)=x^{2}$$

Multiply x and x to get x^{2}.

$$x^{2}f\left(x-5\right)\left(x-1\right)=x^{2}$$

Use the distributive property to multiply x^{2}f by x-5.

$$\left(fx^{3}-5x^{2}f\right)\left(x-1\right)=x^{2}$$

Use the distributive property to multiply fx^{3}-5x^{2}f by x-1 and combine like terms.

$$fx^{4}-6fx^{3}+5fx^{2}=x^{2}$$

Combine all terms containing f.

$$\left(x^{4}-6x^{3}+5x^{2}\right)f=x^{2}$$

Divide both sides by x^{4}-6x^{3}+5x^{2}.

$$\frac{\left(x^{4}-6x^{3}+5x^{2}\right)f}{x^{4}-6x^{3}+5x^{2}}=\frac{x^{2}}{x^{4}-6x^{3}+5x^{2}}$$

Dividing by x^{4}-6x^{3}+5x^{2} undoes the multiplication by x^{4}-6x^{3}+5x^{2}.

$$f=\frac{x^{2}}{x^{4}-6x^{3}+5x^{2}}$$

Divide x^{2} by x^{4}-6x^{3}+5x^{2}.

$$f=\frac{1}{\left(x-5\right)\left(x-1\right)}$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{4f^{2}+f}}{f}+3\text{; }x=-\frac{\sqrt{4f^{2}+f}}{f}+3\text{, }&f>0\text{ or }f\leq -\frac{1}{4}\end{matrix}\right.$$