Solve for $\tau$ where variable inside square root.

Question

How do you solve for $\tau$?

$$t = \tau + \frac{\sqrt{A^2+v^2\tau^2}}{c} $$

it might be easy but I just can not see how. Thanks.

Answer 1

$$ t=\tau+\frac{\sqrt{A^2 + v^2 \tau^2}}{c}\Longrightarrow t-\tau = \frac{\sqrt{A^2 + v^2 \tau^2}}{c}\Longrightarrow(t-\tau)^2 = \frac{A^2+v^2\tau^2}{c^2} $$ Then solve the resulting quadratic. And as always, when squaring both sides, check to make sure your solution solves the original equation.