Solve M=frac{M_{0}}{sqrt{1-v^2/c^2}}

$$M = \frac { M _ { 0 } } { \sqrt { 1 - \frac { v ^ { 2 } } { c ^ { 2 } } } }$$

Solve for M_0 (complex solution)

$$M_{0}=\sqrt{-\left(\frac{v}{c}\right)^{2}+1}M$$

$$v\neq -c\text{ and }v\neq c\text{ and }c\neq 0$$

Steps for Solving Linear Equation

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{c^{2}}{c^{2}}.

$$M=\frac{M_{0}}{\sqrt{\frac{c^{2}}{c^{2}}-\frac{v^{2}}{c^{2}}}}$$

Since \frac{c^{2}}{c^{2}} and \frac{v^{2}}{c^{2}} have the same denominator, subtract them by subtracting their numerators.

$$M=\frac{M_{0}}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}$$

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

$$\frac{M_{0}}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}=M$$

The equation is in standard form.

$$\frac{1}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}M_{0}=M$$

Divide both sides by \left(\sqrt{\left(c^{2}-v^{2}\right)c^{-2}}\right)^{-1}.

$$\frac{\frac{1}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}M_{0}\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}=\frac{M\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}$$

Dividing by \left(\sqrt{\left(c^{2}-v^{2}\right)c^{-2}}\right)^{-1} undoes the multiplication by \left(\sqrt{\left(c^{2}-v^{2}\right)c^{-2}}\right)^{-1}.

$$M_{0}=\frac{M\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}$$

Divide M by \left(\sqrt{\left(c^{2}-v^{2}\right)c^{-2}}\right)^{-1}.

$$M_{0}=\sqrt{-\frac{v^{2}}{c^{2}}+1}M$$

Solve for M_0

$$M_{0}=\frac{M\sqrt{c^{2}-v^{2}}}{|c|}$$

$$\left(v<c\text{ and }v>-c\text{ and }c\neq 0\text{ and }|v|<|c|\right)\text{ or }\left(v<-c\text{ and }v>c\text{ and }c\neq 0\text{ and }|v|<|c|\right)$$

Steps for Solving Linear Equation

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{c^{2}}{c^{2}}.

$$M=\frac{M_{0}}{\sqrt{\frac{c^{2}}{c^{2}}-\frac{v^{2}}{c^{2}}}}$$

Since \frac{c^{2}}{c^{2}} and \frac{v^{2}}{c^{2}} have the same denominator, subtract them by subtracting their numerators.

$$M=\frac{M_{0}}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}$$

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

$$\frac{M_{0}}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}=M$$

The equation is in standard form.

$$\frac{1}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}M_{0}=M$$

Divide both sides by \left(\sqrt{\left(c^{2}-v^{2}\right)c^{-2}}\right)^{-1}.

$$\frac{\frac{1}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}M_{0}\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}=\frac{M\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}$$

Dividing by \left(\sqrt{\left(c^{2}-v^{2}\right)c^{-2}}\right)^{-1} undoes the multiplication by \left(\sqrt{\left(c^{2}-v^{2}\right)c^{-2}}\right)^{-1}.

$$M_{0}=\frac{M\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}$$

Divide M by \left(\sqrt{\left(c^{2}-v^{2}\right)c^{-2}}\right)^{-1}.

$$M_{0}=\frac{M\sqrt{\left(c-v\right)\left(v+c\right)}}{|c|}$$

Solve for M (complex solution)

$$M=\left(-\left(\frac{v}{c}\right)^{2}+1\right)^{-\frac{1}{2}}M_{0}$$

$$v\neq -c\text{ and }v\neq c\text{ and }c\neq 0$$

Solve for M

$$M=\frac{M_{0}|c|}{\sqrt{c^{2}-v^{2}}}$$

$$\left(|c|>|v|\text{ and }v<c\text{ and }v>-c\text{ and }c\neq 0\right)\text{ or }\left(|c|>|v|\text{ and }v<-c\text{ and }v>c\text{ and }c\neq 0\right)$$