Solve f_{m}(x)=x^2-2(m-1)x+m,m

$$f _ { m } ( x ) = x ^ { 2 } - 2 ( m - 1 ) x + m , m$$

Solve for f_m (complex solution)

$$\left\{\begin{matrix}f_{m}=x-2m+\frac{m}{x}+2\text{, }&x\neq 0\\f_{m}\in \mathrm{C}\text{, }&x=0\text{ and }m=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply -1 and 2 to get -2.

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

Use the distributive property to multiply -2 by m-1.

$$f_{m}x=x^{2}+\left(-2m+2\right)x+m$$

Use the distributive property to multiply -2m+2 by x.

$$f_{m}x=x^{2}-2mx+2x+m$$

The equation is in standard form.

$$xf_{m}=x^{2}-2mx+2x+m$$

Divide both sides by x.

$$\frac{xf_{m}}{x}=\frac{x^{2}-2mx+2x+m}{x}$$

Dividing by x undoes the multiplication by x.

$$f_{m}=\frac{x^{2}-2mx+2x+m}{x}$$

Divide x^{2}-2mx+2x+m by x.

$$f_{m}=x-2m+\frac{m}{x}+2$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{x\left(x-f_{m}+2\right)}{1-2x}\text{, }&x\neq \frac{1}{2}\\m\in \mathrm{C}\text{, }&f_{m}=\frac{5}{2}\text{ and }x=\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

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

Use the distributive property to multiply -2 by m-1.

$$x^{2}+\left(-2m+2\right)x+m=f_{m}x$$

Use the distributive property to multiply -2m+2 by x.

$$x^{2}-2mx+2x+m=f_{m}x$$

Subtract x^{2} from both sides.

$$-2mx+2x+m=f_{m}x-x^{2}$$

Subtract 2x from both sides.

$$-2mx+m=f_{m}x-x^{2}-2x$$

Combine all terms containing m.

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

The equation is in standard form.

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

Divide both sides by -2x+1.

$$\frac{\left(1-2x\right)m}{1-2x}=\frac{x\left(-x+f_{m}-2\right)}{1-2x}$$

Dividing by -2x+1 undoes the multiplication by -2x+1.

$$m=\frac{x\left(-x+f_{m}-2\right)}{1-2x}$$

Solve for f_m

$$\left\{\begin{matrix}f_{m}=x-2m+\frac{m}{x}+2\text{, }&x\neq 0\\f_{m}\in \mathrm{R}\text{, }&x=0\text{ and }m=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply -1 and 2 to get -2.

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

Use the distributive property to multiply -2 by m-1.

$$f_{m}x=x^{2}+\left(-2m+2\right)x+m$$

Use the distributive property to multiply -2m+2 by x.

$$f_{m}x=x^{2}-2mx+2x+m$$

The equation is in standard form.

$$xf_{m}=x^{2}-2mx+2x+m$$

Divide both sides by x.

$$\frac{xf_{m}}{x}=\frac{x^{2}-2mx+2x+m}{x}$$

Dividing by x undoes the multiplication by x.

$$f_{m}=\frac{x^{2}-2mx+2x+m}{x}$$

Divide x^{2}-2mx+2x+m by x.

$$f_{m}=x-2m+\frac{m}{x}+2$$

Solve for m

$$\left\{\begin{matrix}m=-\frac{x\left(x-f_{m}+2\right)}{1-2x}\text{, }&x\neq \frac{1}{2}\\m\in \mathrm{R}\text{, }&f_{m}=\frac{5}{2}\text{ and }x=\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

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

Use the distributive property to multiply -2 by m-1.

$$x^{2}+\left(-2m+2\right)x+m=f_{m}x$$

Use the distributive property to multiply -2m+2 by x.

$$x^{2}-2mx+2x+m=f_{m}x$$

Subtract x^{2} from both sides.

$$-2mx+2x+m=f_{m}x-x^{2}$$

Subtract 2x from both sides.

$$-2mx+m=f_{m}x-x^{2}-2x$$

Combine all terms containing m.

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

The equation is in standard form.

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

Divide both sides by 1-2x.

$$\frac{\left(1-2x\right)m}{1-2x}=\frac{x\left(-x+f_{m}-2\right)}{1-2x}$$

Dividing by 1-2x undoes the multiplication by 1-2x.

$$m=\frac{x\left(-x+f_{m}-2\right)}{1-2x}$$