Solve y=(m-3)x-(m-2)^2+5

$$y = ( m - 3 ) x - ( m - 2 ) ^ { 2 } + 5$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{1+4m-m^{2}-y}{m-3}\text{, }&m\neq 3\\x\in \mathrm{C}\text{, }&y=4\text{ and }m=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply m-3 by x.

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

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

$$y=mx-3x-\left(m^{2}-4m+4\right)+5$$

To find the opposite of m^{2}-4m+4, find the opposite of each term.

$$y=mx-3x-m^{2}+4m-4+5$$

Add -4 and 5 to get 1.

$$y=mx-3x-m^{2}+4m+1$$

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

$$mx-3x-m^{2}+4m+1=y$$

Add m^{2} to both sides.

$$mx-3x+4m+1=y+m^{2}$$

Subtract 4m from both sides.

$$mx-3x+1=y+m^{2}-4m$$

Subtract 1 from both sides.

$$mx-3x=y+m^{2}-4m-1$$

Combine all terms containing x.

$$\left(m-3\right)x=y+m^{2}-4m-1$$

Divide both sides by m-3.

$$\frac{\left(m-3\right)x}{m-3}=\frac{y+m^{2}-4m-1}{m-3}$$

Dividing by m-3 undoes the multiplication by m-3.

$$x=\frac{y+m^{2}-4m-1}{m-3}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{1+4m-m^{2}-y}{m-3}\text{, }&m\neq 3\\x\in \mathrm{R}\text{, }&y=4\text{ and }m=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply m-3 by x.

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

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

$$y=mx-3x-\left(m^{2}-4m+4\right)+5$$

To find the opposite of m^{2}-4m+4, find the opposite of each term.

$$y=mx-3x-m^{2}+4m-4+5$$

Add -4 and 5 to get 1.

$$y=mx-3x-m^{2}+4m+1$$

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

$$mx-3x-m^{2}+4m+1=y$$

Add m^{2} to both sides.

$$mx-3x+4m+1=y+m^{2}$$

Subtract 4m from both sides.

$$mx-3x+1=y+m^{2}-4m$$

Subtract 1 from both sides.

$$mx-3x=y+m^{2}-4m-1$$

Combine all terms containing x.

$$\left(m-3\right)x=y+m^{2}-4m-1$$

Divide both sides by m-3.

$$\frac{\left(m-3\right)x}{m-3}=\frac{y+m^{2}-4m-1}{m-3}$$

Dividing by m-3 undoes the multiplication by m-3.

$$x=\frac{y+m^{2}-4m-1}{m-3}$$

Solve for m (complex solution)

$$m=\frac{-\sqrt{x^{2}-4x-4y+20}+x+4}{2}$$

$$m=\frac{\sqrt{x^{2}-4x-4y+20}+x+4}{2}$$

Solve for m

$$m=\frac{-\sqrt{x^{2}-4x-4y+20}+x+4}{2}$$

$$m=\frac{\sqrt{x^{2}-4x-4y+20}+x+4}{2}\text{, }y\leq \frac{x^{2}}{4}-x+5$$