Solve pirh^3-c^2h^2+gv^2=0

$$\pi r h ^ { 3 } - c ^ { 2 } h ^ { 2 } + g v ^ { 2 } = 0$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=-\left(\frac{h}{v}\right)^{2}\left(\pi hr-c^{2}\right)\text{, }&v\neq 0\\g\in \mathrm{C}\text{, }&v=0\text{ and }\left(r=\frac{c^{2}}{\pi h}\text{ or }h=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \pi rh^{3}-c^{2}h^{2} from both sides. Anything subtracted from zero gives its negation.

$$gv^{2}=-\left(\pi rh^{3}-c^{2}h^{2}\right)$$

To find the opposite of \pi rh^{3}-c^{2}h^{2}, find the opposite of each term.

$$gv^{2}=-\pi rh^{3}+c^{2}h^{2}$$

The equation is in standard form.

$$v^{2}g=c^{2}h^{2}-\pi rh^{3}$$

Divide both sides by v^{2}.

$$\frac{v^{2}g}{v^{2}}=\frac{h^{2}\left(c^{2}-\pi hr\right)}{v^{2}}$$

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

$$g=\frac{h^{2}\left(c^{2}-\pi hr\right)}{v^{2}}$$

Solve for g

$$\left\{\begin{matrix}g=-\left(\frac{h}{v}\right)^{2}\left(\pi hr-c^{2}\right)\text{, }&v\neq 0\\g\in \mathrm{R}\text{, }&v=0\text{ and }\left(r=\frac{c^{2}}{\pi h}\text{ or }h=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \pi rh^{3}-c^{2}h^{2} from both sides. Anything subtracted from zero gives its negation.

$$gv^{2}=-\left(\pi rh^{3}-c^{2}h^{2}\right)$$

To find the opposite of \pi rh^{3}-c^{2}h^{2}, find the opposite of each term.

$$gv^{2}=-\pi rh^{3}+c^{2}h^{2}$$

The equation is in standard form.

$$v^{2}g=c^{2}h^{2}-\pi rh^{3}$$

Divide both sides by v^{2}.

$$\frac{v^{2}g}{v^{2}}=\frac{h^{2}\left(c^{2}-\pi hr\right)}{v^{2}}$$

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

$$g=\frac{h^{2}\left(c^{2}-\pi hr\right)}{v^{2}}$$

Solve for c (complex solution)

$$\left\{\begin{matrix}c=-\sqrt{\pi hr+\left(\frac{v}{h}\right)^{2}g}\text{; }c=\sqrt{\pi hr+\left(\frac{v}{h}\right)^{2}g}\text{, }&h\neq 0\\c\in \mathrm{C}\text{, }&\left(g=0\text{ or }v=0\right)\text{ and }h=0\end{matrix}\right.$$