Solve yz^2+3z-6=0

$$y z ^ { 2 } + 3 z - 6 = 0$$

Solve for y

$$y=-\frac{3\left(z-2\right)}{z^{2}}$$

$$z\neq 0$$

Steps for Solving Linear Equation

Subtract 3z from both sides. Anything subtracted from zero gives its negation.

$$yz^{2}-6=-3z$$

Add 6 to both sides.

$$yz^{2}=-3z+6$$

The equation is in standard form.

$$z^{2}y=6-3z$$

Divide both sides by z^{2}.

$$\frac{z^{2}y}{z^{2}}=\frac{6-3z}{z^{2}}$$

Dividing by z^{2} undoes the multiplication by z^{2}.

$$y=\frac{6-3z}{z^{2}}$$

Divide -3z+6 by z^{2}.

$$y=\frac{3\left(2-z\right)}{z^{2}}$$

Solve for z (complex solution)

$$\left\{\begin{matrix}z=\frac{\sqrt{3\left(8y+3\right)}-3}{2y}\text{; }z=-\frac{\sqrt{3}\left(\sqrt{8y+3}+\sqrt{3}\right)}{2y}\text{, }&y\neq 0\\z=2\text{, }&y=0\end{matrix}\right.$$

Solve for z

$$\left\{\begin{matrix}z=\frac{\sqrt{3\left(8y+3\right)}-3}{2y}\text{; }z=-\frac{\sqrt{3}\left(\sqrt{8y+3}+\sqrt{3}\right)}{2y}\text{, }&y\neq 0\text{ and }y\geq -\frac{3}{8}\\z=2\text{, }&y=0\end{matrix}\right.$$