Solve mg=1/2rhosV^2C

$$m g = \frac { 1 } { 2 } \rho s V ^ { 2 } C$$

Solve for C (complex solution)

$$\left\{\begin{matrix}C=\frac{2gm}{s\rho V^{2}}\text{, }&V\neq 0\text{ and }s\neq 0\text{ and }\rho \neq 0\\C\in \mathrm{C}\text{, }&\left(m=0\text{ and }\rho =0\right)\text{ or }\left(m=0\text{ and }s=0\right)\text{ or }\left(m=0\text{ and }V=0\right)\text{ or }\left(g=0\text{ and }\rho =0\right)\text{ or }\left(g=0\text{ and }s=0\right)\text{ or }\left(g=0\text{ and }V=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\frac{1}{2}\rho sV^{2}C=mg$$

The equation is in standard form.

$$\frac{s\rho V^{2}}{2}C=gm$$

Divide both sides by \frac{1}{2}\rho sV^{2}.

$$\frac{2\times \frac{s\rho V^{2}}{2}C}{s\rho V^{2}}=\frac{2gm}{s\rho V^{2}}$$

Dividing by \frac{1}{2}\rho sV^{2} undoes the multiplication by \frac{1}{2}\rho sV^{2}.

$$C=\frac{2gm}{s\rho V^{2}}$$

Solve for C

$$\left\{\begin{matrix}C=\frac{2gm}{s\rho V^{2}}\text{, }&V\neq 0\text{ and }s\neq 0\text{ and }\rho \neq 0\\C\in \mathrm{R}\text{, }&\left(m=0\text{ and }\rho =0\right)\text{ or }\left(m=0\text{ and }s=0\right)\text{ or }\left(m=0\text{ and }V=0\right)\text{ or }\left(g=0\text{ and }\rho =0\right)\text{ or }\left(g=0\text{ and }s=0\right)\text{ or }\left(g=0\text{ and }V=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\frac{1}{2}\rho sV^{2}C=mg$$

The equation is in standard form.

$$\frac{s\rho V^{2}}{2}C=gm$$

Divide both sides by \frac{1}{2}\rho sV^{2}.

$$\frac{2\times \frac{s\rho V^{2}}{2}C}{s\rho V^{2}}=\frac{2gm}{s\rho V^{2}}$$

Dividing by \frac{1}{2}\rho sV^{2} undoes the multiplication by \frac{1}{2}\rho sV^{2}.

$$C=\frac{2gm}{s\rho V^{2}}$$

Solve for V (complex solution)

$$\left\{\begin{matrix}V=-C^{-\frac{1}{2}}s^{-\frac{1}{2}}\rho ^{-\frac{1}{2}}\sqrt{g}\sqrt{2m}\text{; }V=C^{-\frac{1}{2}}s^{-\frac{1}{2}}\rho ^{-\frac{1}{2}}\sqrt{g}\sqrt{2m}\text{, }&C\neq 0\text{ and }s\neq 0\text{ and }\rho \neq 0\\V\in \mathrm{C}\text{, }&\left(m=0\text{ and }\rho =0\right)\text{ or }\left(m=0\text{ and }s=0\right)\text{ or }\left(m=0\text{ and }C=0\right)\text{ or }\left(g=0\text{ and }\rho =0\right)\text{ or }\left(g=0\text{ and }s=0\right)\text{ or }\left(g=0\text{ and }C=0\right)\end{matrix}\right.$$