Why is a negative sign included in Equation (6) for central-force motion?

[sams]

In Chapter 8: Central-Force Motion, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 323, Problem 8-5, we are asked to show that the two particles will collide after a time $\tau/4√2$.

I don't have any problems with the derivations and with the integrations, but I want to know please why the authors put a negative sign in Equation (6) and what do they mean that the negative sign was included due to the fact that the time increases as the distance decreases?

Thanks a lot for your help...

 

[PeroK]

In my opinion the negative sign should have appeared in equation (5). Equation (4) for $\dot{x}^2$ has two solutions. In this case, the negative square root is required as $\dot{x}$ is negative, for the physical reason given

 

[sams]

Thank you so much @PeroK for your continuous support...