Solve x^2+y^2+(k-2)x-2ky-k-4=0

$$x ^ { 2 } + y ^ { 2 } + ( k - 2 ) x - 2 k y - k - 4 = 0$$

Solve for k (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply k-2 by x.

$$x^{2}+y^{2}+kx-2x-2ky-k-4=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$y^{2}+kx-2x-2ky-k-4=-x^{2}$$

Subtract y^{2} from both sides.

$$kx-2x-2ky-k-4=-x^{2}-y^{2}$$

Add 2x to both sides.

$$kx-2ky-k-4=-x^{2}-y^{2}+2x$$

Add 4 to both sides.

$$kx-2ky-k=-x^{2}-y^{2}+2x+4$$

Combine all terms containing k.

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

The equation is in standard form.

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

Divide both sides by x-2y-1.

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

Dividing by x-2y-1 undoes the multiplication by x-2y-1.

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

Solve for k

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

Steps for Solving Linear Equation

Use the distributive property to multiply k-2 by x.

$$x^{2}+y^{2}+kx-2x-2ky-k-4=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$y^{2}+kx-2x-2ky-k-4=-x^{2}$$

Subtract y^{2} from both sides.

$$kx-2x-2ky-k-4=-x^{2}-y^{2}$$

Add 2x to both sides.

$$kx-2ky-k-4=-x^{2}-y^{2}+2x$$

Add 4 to both sides.

$$kx-2ky-k=-x^{2}-y^{2}+2x+4$$

Combine all terms containing k.

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

The equation is in standard form.

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

Divide both sides by x-2y-1.

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

Dividing by x-2y-1 undoes the multiplication by x-2y-1.

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

Solve for x (complex solution)

$$x=\frac{\sqrt{20+k^{2}+8ky-4y^{2}}}{2}-\frac{k}{2}+1$$

$$x=-\frac{\sqrt{20+k^{2}+8ky-4y^{2}}}{2}-\frac{k}{2}+1$$

Solve for x

$$x=\frac{\sqrt{20+k^{2}+8ky-4y^{2}}}{2}-\frac{k}{2}+1$$

$$x=-\frac{\sqrt{20+k^{2}+8ky-4y^{2}}}{2}-\frac{k}{2}+1\text{, }y\geq -\frac{\sqrt{80k^{2}+320}}{8}+k\text{ and }y\leq \frac{\sqrt{80k^{2}+320}}{8}+k$$