Solve fx=xsqrt{left(x-1right)^2}

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

Solve for f (complex solution)

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

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$fx=x\sqrt{x^{2}-2x+1}$$

The equation is in standard form.

$$xf=x\sqrt{x^{2}-2x+1}$$

Divide both sides by x.

$$\frac{xf}{x}=\frac{x\sqrt{\left(x-1\right)^{2}}}{x}$$

Dividing by x undoes the multiplication by x.

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

Divide x\sqrt{\left(x-1\right)^{2}} by x.

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

Solve for f

$$\left\{\begin{matrix}\\f=|x-1|\text{, }&\text{unconditionally}\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$fx=x\sqrt{x^{2}-2x+1}$$

The equation is in standard form.

$$xf=x\sqrt{x^{2}-2x+1}$$

Divide both sides by x.

$$\frac{xf}{x}=\frac{x|x-1|}{x}$$

Dividing by x undoes the multiplication by x.

$$f=\frac{x|x-1|}{x}$$

Divide x|x-1| by x.

$$f=|x-1|$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=1-f\text{; }x=f+1\text{, }&arg(f)<\pi \text{ or }f=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=1-f\text{; }x=f+1\text{, }&f\geq 0\end{matrix}\right.$$