Solve R_0^2*r-2R_{0}r^2+r^3=GMt^2/2

$$R _ { 0 } ^ { 2 } \cdot r - 2 R _ { 0 } r ^ { 2 } + r ^ { 3 } = \frac { G M t ^ { 2 } } { 2 }$$

Solve for G (complex solution)

$$\left\{\begin{matrix}G=\frac{2\times \left(\frac{R_{0}-r}{t}\right)^{2}r}{M}\text{, }&t\neq 0\text{ and }M\neq 0\\G\in \mathrm{C}\text{, }&\left(R_{0}=r\text{ and }M=0\right)\text{ or }\left(R_{0}=r\text{ and }t=0\right)\text{ or }\left(r=0\text{ and }M=0\right)\text{ or }\left(r=0\text{ and }t=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$2\left(R_{0}^{2}r-2R_{0}r^{2}\right)+2r^{3}=GMt^{2}$$

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

$$GMt^{2}=2\left(R_{0}^{2}r-2R_{0}r^{2}\right)+2r^{3}$$

Use the distributive property to multiply 2 by R_{0}^{2}r-2R_{0}r^{2}.

$$GMt^{2}=2R_{0}^{2}r-4R_{0}r^{2}+2r^{3}$$

The equation is in standard form.

$$Mt^{2}G=2r^{3}-4R_{0}r^{2}+2rR_{0}^{2}$$

Divide both sides by Mt^{2}.

$$\frac{Mt^{2}G}{Mt^{2}}=\frac{2r\left(R_{0}-r\right)^{2}}{Mt^{2}}$$

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

$$G=\frac{2r\left(R_{0}-r\right)^{2}}{Mt^{2}}$$

Solve for M (complex solution)

$$\left\{\begin{matrix}M=\frac{2\times \left(\frac{R_{0}-r}{t}\right)^{2}r}{G}\text{, }&t\neq 0\text{ and }G\neq 0\\M\in \mathrm{C}\text{, }&\left(R_{0}=r\text{ and }G=0\right)\text{ or }\left(R_{0}=r\text{ and }t=0\right)\text{ or }\left(r=0\text{ and }G=0\right)\text{ or }\left(r=0\text{ and }t=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$2\left(R_{0}^{2}r-2R_{0}r^{2}\right)+2r^{3}=GMt^{2}$$

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

$$GMt^{2}=2\left(R_{0}^{2}r-2R_{0}r^{2}\right)+2r^{3}$$

Use the distributive property to multiply 2 by R_{0}^{2}r-2R_{0}r^{2}.

$$GMt^{2}=2R_{0}^{2}r-4R_{0}r^{2}+2r^{3}$$

The equation is in standard form.

$$Gt^{2}M=2r^{3}-4R_{0}r^{2}+2rR_{0}^{2}$$

Divide both sides by Gt^{2}.

$$\frac{Gt^{2}M}{Gt^{2}}=\frac{2r\left(R_{0}-r\right)^{2}}{Gt^{2}}$$

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

$$M=\frac{2r\left(R_{0}-r\right)^{2}}{Gt^{2}}$$

Solve for G

$$\left\{\begin{matrix}G=\frac{2\times \left(\frac{R_{0}-r}{t}\right)^{2}r}{M}\text{, }&t\neq 0\text{ and }M\neq 0\\G\in \mathrm{R}\text{, }&\left(R_{0}=r\text{ and }M=0\right)\text{ or }\left(R_{0}=r\text{ and }t=0\right)\text{ or }\left(r=0\text{ and }M=0\right)\text{ or }\left(r=0\text{ and }t=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$2\left(R_{0}^{2}r-2R_{0}r^{2}\right)+2r^{3}=GMt^{2}$$

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

$$GMt^{2}=2\left(R_{0}^{2}r-2R_{0}r^{2}\right)+2r^{3}$$

Use the distributive property to multiply 2 by R_{0}^{2}r-2R_{0}r^{2}.

$$GMt^{2}=2R_{0}^{2}r-4R_{0}r^{2}+2r^{3}$$

The equation is in standard form.

$$Mt^{2}G=2r^{3}-4R_{0}r^{2}+2rR_{0}^{2}$$

Divide both sides by Mt^{2}.

$$\frac{Mt^{2}G}{Mt^{2}}=\frac{2r\left(R_{0}-r\right)^{2}}{Mt^{2}}$$

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

$$G=\frac{2r\left(R_{0}-r\right)^{2}}{Mt^{2}}$$

Solve for M

$$\left\{\begin{matrix}M=\frac{2\times \left(\frac{R_{0}-r}{t}\right)^{2}r}{G}\text{, }&t\neq 0\text{ and }G\neq 0\\M\in \mathrm{R}\text{, }&\left(R_{0}=r\text{ and }G=0\right)\text{ or }\left(R_{0}=r\text{ and }t=0\right)\text{ or }\left(r=0\text{ and }G=0\right)\text{ or }\left(r=0\text{ and }t=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$2\left(R_{0}^{2}r-2R_{0}r^{2}\right)+2r^{3}=GMt^{2}$$

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

$$GMt^{2}=2\left(R_{0}^{2}r-2R_{0}r^{2}\right)+2r^{3}$$

Use the distributive property to multiply 2 by R_{0}^{2}r-2R_{0}r^{2}.

$$GMt^{2}=2R_{0}^{2}r-4R_{0}r^{2}+2r^{3}$$

The equation is in standard form.

$$Gt^{2}M=2r^{3}-4R_{0}r^{2}+2rR_{0}^{2}$$

Divide both sides by Gt^{2}.

$$\frac{Gt^{2}M}{Gt^{2}}=\frac{2r\left(R_{0}-r\right)^{2}}{Gt^{2}}$$

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

$$M=\frac{2r\left(R_{0}-r\right)^{2}}{Gt^{2}}$$