Solve y=x^2-4x+mx+3

$$y = x ^ { 2 } - 4 x + m x + 3$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=\frac{-x^{2}+4x+y-3}{x}\text{, }&x\neq 0\\m\in \mathrm{C}\text{, }&y=3\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{2}-4x+mx+3=y$$

Subtract x^{2} from both sides.

$$-4x+mx+3=y-x^{2}$$

Add 4x to both sides.

$$mx+3=y-x^{2}+4x$$

Subtract 3 from both sides.

$$mx=y-x^{2}+4x-3$$

The equation is in standard form.

$$xm=-x^{2}+4x+y-3$$

Divide both sides by x.

$$\frac{xm}{x}=\frac{-x^{2}+4x+y-3}{x}$$

Dividing by x undoes the multiplication by x.

$$m=\frac{-x^{2}+4x+y-3}{x}$$

Divide y-x^{2}+4x-3 by x.

$$m=-x+\frac{y-3}{x}+4$$

Solve for m

$$\left\{\begin{matrix}m=\frac{-x^{2}+4x+y-3}{x}\text{, }&x\neq 0\\m\in \mathrm{R}\text{, }&y=3\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{2}-4x+mx+3=y$$

Subtract x^{2} from both sides.

$$-4x+mx+3=y-x^{2}$$

Add 4x to both sides.

$$mx+3=y-x^{2}+4x$$

Subtract 3 from both sides.

$$mx=y-x^{2}+4x-3$$

The equation is in standard form.

$$xm=-x^{2}+4x+y-3$$

Divide both sides by x.

$$\frac{xm}{x}=\frac{-x^{2}+4x+y-3}{x}$$

Dividing by x undoes the multiplication by x.

$$m=\frac{-x^{2}+4x+y-3}{x}$$

Divide y-x^{2}+4x-3 by x.

$$m=-x+\frac{y-3}{x}+4$$

Solve for x (complex solution)

$$x=\frac{\sqrt{4y+m^{2}-8m+4}}{2}-\frac{m}{2}+2$$

$$x=-\frac{\sqrt{4y+m^{2}-8m+4}}{2}-\frac{m}{2}+2$$

Solve for x

$$x=\frac{\sqrt{4y+m^{2}-8m+4}}{2}-\frac{m}{2}+2$$

$$x=-\frac{\sqrt{4y+m^{2}-8m+4}}{2}-\frac{m}{2}+2\text{, }y\geq -\frac{m^{2}}{4}+2m-1$$