Solve S=sqrt{30D*f*n}

$$S = \sqrt { 30 D \cdot f \cdot n }$$

Solve for D

$$\left\{\begin{matrix}D=\frac{S^{2}}{30fn}\text{, }&S\geq 0\text{ and }f\neq 0\text{ and }n\neq 0\\D\in \mathrm{R}\text{, }&\left(f=0\text{ or }n=0\right)\text{ and }S=0\end{matrix}\right.$$

Steps for Solving Radical Equations

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

$$\sqrt{30Dfn}=S$$

Square both sides of the equation.

$$30fnD=S^{2}$$

Divide both sides by 30fn.

$$\frac{30fnD}{30fn}=\frac{S^{2}}{30fn}$$

Dividing by 30fn undoes the multiplication by 30fn.

$$D=\frac{S^{2}}{30fn}$$

Solve for D (complex solution)

$$\left\{\begin{matrix}D=\frac{S^{2}}{30fn}\text{, }&n\neq 0\text{ and }f\neq 0\text{ and }\left(S=0\text{ or }arg(S)<\pi \right)\\D\in \mathrm{C}\text{, }&\left(f=0\text{ or }n=0\right)\text{ and }S=0\end{matrix}\right.$$

Solve for S (complex solution)

$$S=\sqrt{30Dfn}$$

Solve for S

$$S=\sqrt{30Dfn}$$

$$\left(n\leq 0\text{ and }f\leq 0\text{ and }D\geq 0\right)\text{ or }\left(f\geq 0\text{ and }n\geq 0\text{ and }D\geq 0\right)\text{ or }\left(D\leq 0\text{ and }f\leq 0\text{ and }n\geq 0\right)\text{ or }\left(D\leq 0\text{ and }n\leq 0\text{ and }f\geq 0\right)$$