Solve t=T/sqrt{1-frac{v^2{c^2}}}

$$t = \frac { T } { \sqrt { 1 - \frac { v ^ { 2 } } { c ^ { 2 } } } }$$

Solve for T (complex solution)

$$T=\sqrt{-\left(\frac{v}{c}\right)^{2}+1}t$$

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

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}}.

$$t=\frac{T}{\sqrt{\frac{c^{2}}{c^{2}}-\frac{v^{2}}{c^{2}}}}$$

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

$$t=\frac{T}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}$$

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

$$\frac{T}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}=t$$

The equation is in standard form.

$$\frac{1}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}T=t$$

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

$$\frac{\frac{1}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}T\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}=\frac{t\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}$$

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}.

$$T=\frac{t\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}$$

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

$$T=\sqrt{-\frac{v^{2}}{c^{2}}+1}t$$

Solve for T

$$T=\frac{t\sqrt{c^{2}-v^{2}}}{|c|}$$

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

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}}.

$$t=\frac{T}{\sqrt{\frac{c^{2}}{c^{2}}-\frac{v^{2}}{c^{2}}}}$$

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

$$t=\frac{T}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}$$

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

$$\frac{T}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}=t$$

The equation is in standard form.

$$\frac{1}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}T=t$$

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

$$\frac{\frac{1}{\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}T\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}=\frac{t\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}$$

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}.

$$T=\frac{t\sqrt{\frac{c^{2}-v^{2}}{c^{2}}}}{1}$$

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

$$T=\frac{t\sqrt{\left(c-v\right)\left(v+c\right)}}{|c|}$$