Solve xm+3divxm+1=x_{2}+3x-5

$$x m + 3 \div x m + 1 = x _ { 2 } + 3 x - 5$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{x\left(6-x_{2}-3x\right)}{x^{2}+3}\text{, }&x\neq -\sqrt{3}i\text{ and }x\neq \sqrt{3}i\text{ and }x\neq 0\\m\in \mathrm{C}\text{, }&\left(x=\sqrt{3}i\text{ and }x_{2}=-3\sqrt{3}i+6\right)\text{ or }\left(x=-\sqrt{3}i\text{ and }x_{2}=6+3\sqrt{3}i\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$xmx+3m+x=xx_{2}+3xx+x\left(-5\right)$$

Multiply x and x to get x^{2}.

$$x^{2}m+3m+x=xx_{2}+3xx+x\left(-5\right)$$

Multiply x and x to get x^{2}.

$$x^{2}m+3m+x=xx_{2}+3x^{2}+x\left(-5\right)$$

Subtract x from both sides.

$$x^{2}m+3m=xx_{2}+3x^{2}+x\left(-5\right)-x$$

Combine x\left(-5\right) and -x to get -6x.

$$x^{2}m+3m=xx_{2}+3x^{2}-6x$$

Combine all terms containing m.

$$\left(x^{2}+3\right)m=xx_{2}+3x^{2}-6x$$

The equation is in standard form.

$$\left(x^{2}+3\right)m=3x^{2}+xx_{2}-6x$$

Divide both sides by x^{2}+3.

$$\frac{\left(x^{2}+3\right)m}{x^{2}+3}=\frac{x\left(3x+x_{2}-6\right)}{x^{2}+3}$$

Dividing by x^{2}+3 undoes the multiplication by x^{2}+3.

$$m=\frac{x\left(3x+x_{2}-6\right)}{x^{2}+3}$$

Solve for m

$$m=-\frac{x\left(6-x_{2}-3x\right)}{x^{2}+3}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$xmx+3m+x=xx_{2}+3xx+x\left(-5\right)$$

Multiply x and x to get x^{2}.

$$x^{2}m+3m+x=xx_{2}+3xx+x\left(-5\right)$$

Multiply x and x to get x^{2}.

$$x^{2}m+3m+x=xx_{2}+3x^{2}+x\left(-5\right)$$

Subtract x from both sides.

$$x^{2}m+3m=xx_{2}+3x^{2}+x\left(-5\right)-x$$

Combine x\left(-5\right) and -x to get -6x.

$$x^{2}m+3m=xx_{2}+3x^{2}-6x$$

Combine all terms containing m.

$$\left(x^{2}+3\right)m=xx_{2}+3x^{2}-6x$$

The equation is in standard form.

$$\left(x^{2}+3\right)m=3x^{2}+xx_{2}-6x$$

Divide both sides by x^{2}+3.

$$\frac{\left(x^{2}+3\right)m}{x^{2}+3}=\frac{x\left(3x+x_{2}-6\right)}{x^{2}+3}$$

Dividing by x^{2}+3 undoes the multiplication by x^{2}+3.

$$m=\frac{x\left(3x+x_{2}-6\right)}{x^{2}+3}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{x_{2}^{2}-12x_{2}-12m^{2}+36m+36}+x_{2}-6}{2\left(m-3\right)}\text{, }&\left(m\neq 3\text{ and }m\neq 0\right)\text{ or }\left(m\neq 3\text{ and }arg(6-x_{2})<\pi \text{ and }x_{2}\neq 6\right)\\x=\frac{\sqrt{x_{2}^{2}-12x_{2}-12m^{2}+36m+36}+x_{2}-6}{2\left(m-3\right)}\text{, }&\left(m\neq 3\text{ and }m\neq 0\right)\text{ or }\left(m\neq 3\text{ and }arg(6-x_{2})\geq \pi \text{ and }x_{2}\neq 6\right)\\x=-\frac{9}{6-x_{2}}\text{, }&m=3\text{ and }x_{2}\neq 6\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{x_{2}^{2}-12x_{2}-12m^{2}+36m+36}+x_{2}-6}{2\left(m-3\right)}\text{, }&\left(-\left(6-x_{2}\right)<0\text{ or }m\neq 0\right)\text{ and }m\neq 3\text{ and }m\geq -\frac{\sqrt{48\left(-\left(6-x_{2}\right)\right)^{2}+1296}}{24}+\frac{3}{2}\text{ and }m\leq \frac{\sqrt{48\left(-\left(6-x_{2}\right)\right)^{2}+1296}}{24}+\frac{3}{2}\text{ and }\left(x_{2}<6\text{ or }m\neq 0\right)\\x=\frac{\sqrt{x_{2}^{2}-12x_{2}-12m^{2}+36m+36}+x_{2}-6}{2\left(m-3\right)}\text{, }&\left(6-x_{2}<0\text{ or }m\neq 0\right)\text{ and }m\neq 3\text{ and }m\geq -\frac{\sqrt{48\left(6-x_{2}\right)^{2}+1296}}{24}+\frac{3}{2}\text{ and }m\leq \frac{\sqrt{48\left(6-x_{2}\right)^{2}+1296}}{24}+\frac{3}{2}\text{ and }\left(x_{2}>6\text{ or }m\neq 0\right)\\x=-\frac{9}{6-x_{2}}\text{, }&m=3\text{ and }x_{2}\neq 6\end{matrix}\right.$$