Solve fx=x^2-8x+19/x-5

$$fx= \frac{ { x }^{ 2 } -8x+19 }{ x-5 }$$

Solve for f

$$f=-\frac{-x^{2}+8x-19}{x\left(x-5\right)}$$

$$x\neq 5\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x-5.

$$fx\left(x-5\right)=x^{2}-8x+19$$

Use the distributive property to multiply fx by x-5.

$$fx^{2}-5fx=x^{2}-8x+19$$

Combine all terms containing f.

$$\left(x^{2}-5x\right)f=x^{2}-8x+19$$

Divide both sides by x^{2}-5x.

$$\frac{\left(x^{2}-5x\right)f}{x^{2}-5x}=\frac{x^{2}-8x+19}{x^{2}-5x}$$

Dividing by x^{2}-5x undoes the multiplication by x^{2}-5x.

$$f=\frac{x^{2}-8x+19}{x^{2}-5x}$$

Divide x^{2}-8x+19 by x^{2}-5x.

$$f=\frac{x^{2}-8x+19}{x\left(x-5\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{25f^{2}-4f-12}+5f-8}{2\left(f-1\right)}\text{; }x=\frac{-\sqrt{25f^{2}-4f-12}+5f-8}{2\left(f-1\right)}\text{, }&f\neq 1\\x=\frac{19}{3}\text{, }&f=1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{25f^{2}-4f-12}+5f-8}{2\left(f-1\right)}\text{; }x=\frac{-\sqrt{25f^{2}-4f-12}+5f-8}{2\left(f-1\right)}\text{, }&f\leq \frac{2-4\sqrt{19}}{25}\text{ or }\left(f\neq 1\text{ and }f\geq \frac{4\sqrt{19}+2}{25}\right)\\x=\frac{19}{3}\text{, }&f=1\end{matrix}\right.$$