Solve x^2+3x=ux+2u+x

$$x ^ { 2 } + 3 x = u x + 2 u + x$$

Solve for u (complex solution)

$$\left\{\begin{matrix}\\u=x\text{, }&\text{unconditionally}\\u\in \mathrm{C}\text{, }&x=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ux+2u+x=x^{2}+3x$$

Subtract x from both sides.

$$ux+2u=x^{2}+3x-x$$

Combine 3x and -x to get 2x.

$$ux+2u=x^{2}+2x$$

Combine all terms containing u.

$$\left(x+2\right)u=x^{2}+2x$$

Divide both sides by x+2.

$$\frac{\left(x+2\right)u}{x+2}=\frac{x\left(x+2\right)}{x+2}$$

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

$$u=\frac{x\left(x+2\right)}{x+2}$$

Divide x\left(2+x\right) by x+2.

$$u=x$$

Solve for u

$$\left\{\begin{matrix}\\u=x\text{, }&\text{unconditionally}\\u\in \mathrm{R}\text{, }&x=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ux+2u+x=x^{2}+3x$$

Subtract x from both sides.

$$ux+2u=x^{2}+3x-x$$

Combine 3x and -x to get 2x.

$$ux+2u=x^{2}+2x$$

Combine all terms containing u.

$$\left(x+2\right)u=x^{2}+2x$$

Divide both sides by x+2.

$$\frac{\left(x+2\right)u}{x+2}=\frac{x\left(x+2\right)}{x+2}$$

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

$$u=\frac{x\left(x+2\right)}{x+2}$$

Divide x\left(2+x\right) by x+2.

$$u=x$$

Solve for x

$$x=u$$

$$x=-2$$