Solve frac{C_{n}h}{C_{n}}=R^2/(R+H)^2

$$\frac { C _ { n } h } { C _ { n } } = \frac { R ^ { 2 } } { ( R + H ) ^ { 2 } }$$

Solve for C_n (complex solution)

$$C_{n}\neq 0$$

$$\left(R=\frac{-Hh+\sqrt{hH^{2}}}{h-1}\text{ and }H\neq 0\text{ and }h\neq 1\right)\text{ or }\left(R=-\frac{Hh+\sqrt{hH^{2}}}{h-1}\text{ and }h\neq 0\text{ and }H\neq 0\text{ and }h\neq 1\right)\text{ or }\left(R=-\frac{H}{2}\text{ and }h=1\text{ and }H\neq 0\right)\text{ or }\left(h=1\text{ and }H=0\text{ and }R\neq 0\right)$$

Steps for Solving Linear Equation

Variable C_{n} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by C_{n}\left(H+R\right)^{2}, the least common multiple of C_{n},\left(R+H\right)^{2}.

$$\left(H+R\right)^{2}C_{n}h=C_{n}R^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(H+R\right)^{2}.

$$\left(H^{2}+2HR+R^{2}\right)C_{n}h=C_{n}R^{2}$$

Use the distributive property to multiply H^{2}+2HR+R^{2} by C_{n}.

$$\left(H^{2}C_{n}+2HRC_{n}+R^{2}C_{n}\right)h=C_{n}R^{2}$$

Use the distributive property to multiply H^{2}C_{n}+2HRC_{n}+R^{2}C_{n} by h.

$$H^{2}C_{n}h+2HRC_{n}h+R^{2}C_{n}h=C_{n}R^{2}$$

Subtract C_{n}R^{2} from both sides.

$$H^{2}C_{n}h+2HRC_{n}h+R^{2}C_{n}h-C_{n}R^{2}=0$$

Reorder the terms.

$$2C_{n}HRh-C_{n}R^{2}+C_{n}hH^{2}+C_{n}hR^{2}=0$$

Combine all terms containing C_{n}.

$$\left(2HRh-R^{2}+hH^{2}+hR^{2}\right)C_{n}=0$$

Divide 0 by hR^{2}+2hRH+hH^{2}-R^{2}.

$$C_{n}=0$$

Variable C_{n} cannot be equal to 0.

$$C_{n}\in \emptyset$$

Solve for C_n

$$C_{n}\neq 0$$

$$\left(R=0\text{ and }h=0\text{ and }H\neq 0\right)\text{ or }\left(R\neq 0\text{ and }h=1\text{ and }H=0\right)\text{ or }\left(R=-\frac{H}{2}\text{ and }H\neq 0\text{ and }h=1\right)\text{ or }\left(R=\frac{-Hh+\sqrt{h}H}{h-1}\text{ and }H\neq 0\text{ and }h\neq 1\text{ and }h>0\right)\text{ or }\left(R=-\frac{Hh+\sqrt{h}H}{h-1}\text{ and }H\neq 0\text{ and }h\neq 1\text{ and }h>0\right)$$

Steps for Solving Linear Equation

Variable C_{n} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by C_{n}\left(H+R\right)^{2}, the least common multiple of C_{n},\left(R+H\right)^{2}.

$$\left(H+R\right)^{2}C_{n}h=C_{n}R^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(H+R\right)^{2}.

$$\left(H^{2}+2HR+R^{2}\right)C_{n}h=C_{n}R^{2}$$

Use the distributive property to multiply H^{2}+2HR+R^{2} by C_{n}.

$$\left(H^{2}C_{n}+2HRC_{n}+R^{2}C_{n}\right)h=C_{n}R^{2}$$

Use the distributive property to multiply H^{2}C_{n}+2HRC_{n}+R^{2}C_{n} by h.

$$H^{2}C_{n}h+2HRC_{n}h+R^{2}C_{n}h=C_{n}R^{2}$$

Subtract C_{n}R^{2} from both sides.

$$H^{2}C_{n}h+2HRC_{n}h+R^{2}C_{n}h-C_{n}R^{2}=0$$

Reorder the terms.

$$2C_{n}HRh-C_{n}R^{2}+C_{n}hH^{2}+C_{n}hR^{2}=0$$

Combine all terms containing C_{n}.

$$\left(2HRh-R^{2}+hH^{2}+hR^{2}\right)C_{n}=0$$

Divide 0 by hR^{2}+2hRH+hH^{2}-R^{2}.

$$C_{n}=0$$

Variable C_{n} cannot be equal to 0.

$$C_{n}\in \emptyset$$

Solve for H (complex solution)

$$\left\{\begin{matrix}H=-R-h^{-\frac{1}{2}}R\text{; }H=h^{-\frac{1}{2}}R-R\text{, }&R\neq 0\text{ and }h\neq 0\text{ and }C_{n}\neq 0\\H\neq 0\text{, }&h=0\text{ and }R=0\text{ and }C_{n}\neq 0\end{matrix}\right.$$

Solve for H

$$\left\{\begin{matrix}H=-R+\frac{R}{\sqrt{h}}\text{; }H=-R-\frac{R}{\sqrt{h}}\text{, }&R\neq 0\text{ and }h>0\text{ and }C_{n}\neq 0\\H\neq 0\text{, }&h=0\text{ and }R=0\text{ and }C_{n}\neq 0\end{matrix}\right.$$