Solve x^r-fx=6

$$x ^ { r } - f x = 6$$

Solve for f (complex solution)

$$f=\frac{x^{r}-6}{x}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Subtract x^{r} from both sides.

$$-fx=6-x^{r}$$

The equation is in standard form.

$$\left(-x\right)f=6-x^{r}$$

Divide both sides by -x.

$$\frac{\left(-x\right)f}{-x}=\frac{6-x^{r}}{-x}$$

Dividing by -x undoes the multiplication by -x.

$$f=\frac{6-x^{r}}{-x}$$

Divide -x^{r}+6 by -x.

$$f=x^{r-1}-\frac{6}{x}$$

Solve for f

$$f=\frac{x^{r}-6}{x}$$

$$\left(Denominator(r)\text{bmod}2=1\text{ and }x<0\right)\text{ or }x>0$$

Steps for Solving Linear Equation

Subtract x^{r} from both sides.

$$-fx=6-x^{r}$$

The equation is in standard form.

$$\left(-x\right)f=6-x^{r}$$

Divide both sides by -x.

$$\frac{\left(-x\right)f}{-x}=\frac{6-x^{r}}{-x}$$

Dividing by -x undoes the multiplication by -x.

$$f=\frac{6-x^{r}}{-x}$$

Divide 6-x^{r} by -x.

$$f=x^{r-1}-\frac{6}{x}$$

Solve for r (complex solution)

$$\left\{\begin{matrix}r=\log_{x}\left(fx+6\right)+\frac{2\pi n_{1}i}{\ln(x)}\text{, }n_{1}\in \mathrm{Z}\text{, }&\left(f=0\text{ or }x\neq -\frac{6}{f}\right)\text{ and }x\neq 1\text{ and }x\neq 0\\r\in \mathrm{C}\text{, }&f=-5\text{ and }x=1\end{matrix}\right.$$

Solve for r

$$\left\{\begin{matrix}r=\log_{x}\left(fx+6\right)\text{, }&\left(x<-\frac{6}{f}\text{ and }x\neq 1\text{ and }x>0\text{ and }f<0\right)\text{ or }\left(x\neq 1\text{ and }f=0\text{ and }x>0\right)\text{ or }\left(x>-\frac{6}{f}\text{ and }x\neq 1\text{ and }x>0\text{ and }f>0\right)\\r\in \mathrm{R}\text{, }&\left(f=-5\text{ and }x=1\right)\text{ or }\left(f=7\text{ and }x=-1\text{ and }Denominator(r)\text{bmod}2=1\text{ and }Numerator(r)\text{bmod}2=1\right)\end{matrix}\right.$$