t represents time in seconds, and time cannot be negative, so I would assume you reject t=-1 as a solution and take only the other solution.
-1 seconds. That doesn't make any sense.
Doesn't it? Here's a question - what if I want to know how long I have to wait until you reply to my previous post? Well, you already have replied, so I'd have to wait -11 minutes. OR what if I want to know how long until my pizza is ready to eat 5 minutes after it's finished cooking? Well, it was ready 5 minutes ago, so I'd have to wait -5 minutes. And an example maybe a bit more relevant to the question: if I arrive to a bus stop 5 minutes after the bus has, and the next one is in 20 minutes, at what times will a bus arrive? Either 5 minutes ago, or 20 minutes from now.
You both have assumed that because t is time, it cannot be negative - but why not? All we know is that we started the timer when x_1 was at position 2, and x_2 was at position -2. That doesn't mean they were ALWAYS at those positions, that's just the positions at which we started the timer. At no point did the question say that t HAD to be greater than 0 - you guys assumed this.
In fact, if you look at any VCAA exam where t is the time, they will ALWAYS specify a domain for t. Sure, /usually/ that domain is just t>0, and I don't think there's been a case yet where it could be negative (there probably is one in some normal distribution question that I'm missing - tbh I went straight to the specialist exams, because they mention time more), but that doesn't mean it can be negative. Time is just a relative construct, after all.
Potentially another analogy that might help - let's say I'm cooking pizza, it takes 10 minutes to cook, but I don't start the timer until 3 minutes after I put it in the oven. Does that mean that I still take the pizza out when the timer says it's been 10 minutes? Of course not, my pizza will be burnt - I'd take it out at 7 minutes, because -3+10=7. You could even say that I, the person, will position myself at the oven at the times t=-3 and t=7 minutes.
EDIT: Also, fun fact, the file name for the 2018 specialist exam 2 is "2016 Specialist Mathematics Written examination 2" lmfao