Solve x^2+y^2+z^2+x+4y-z=0

$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + x + 4 y - z = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{1+4z-4z^{2}-16y-4y^{2}}-1}{2}$$

$$x=\frac{-\sqrt{1+4z-4z^{2}-16y-4y^{2}}-1}{2}$$

Solve for y (complex solution)

$$y=\sqrt{4+z-z^{2}-x-x^{2}}-2$$

$$y=-\sqrt{4+z-z^{2}-x-x^{2}}-2$$

Solve for x

$$x=\frac{\sqrt{1+4z-4z^{2}-16y-4y^{2}}-1}{2}$$

$$x=\frac{-\sqrt{1+4z-4z^{2}-16y-4y^{2}}-1}{2}\text{, }\left(z>\frac{1-3\sqrt{2}}{2}\text{ and }z<\frac{3\sqrt{2}+1}{2}\text{ and }y\leq \frac{\sqrt{17+4z-4z^{2}}}{2}-2\text{ and }y\geq -\frac{\sqrt{17+4z-4z^{2}}}{2}-2\right)\text{ or }\left(z\geq \frac{1-3\sqrt{2}}{2}\text{ and }z\leq \frac{3\sqrt{2}+1}{2}\text{ and }y=-2\right)$$

Solve for y

$$y=\sqrt{4+z-z^{2}-x-x^{2}}-2$$

$$y=-\sqrt{4+z-z^{2}-x-x^{2}}-2\text{, }\left(z>\frac{1-3\sqrt{2}}{2}\text{ and }z<\frac{3\sqrt{2}+1}{2}\text{ and }x\leq \frac{\sqrt{17+4z-4z^{2}}-1}{2}\text{ and }x\geq \frac{-\sqrt{17+4z-4z^{2}}-1}{2}\right)\text{ or }\left(z\geq \frac{1-3\sqrt{2}}{2}\text{ and }z\leq \frac{3\sqrt{2}+1}{2}\text{ and }x=-\frac{1}{2}\right)$$