1. The problem statement, all variables and given/known data
Vroom, vroom! As soon as a traffic light turns green, a car
speeds up from rest to 50.0 mi/h with constant acceleration
9.00 mi/h + s. In the adjoining bike lane, a cyclist
speeds up from rest to 20.0 mi/h with constant acceleration
13.0 mi/h + s. Each vehicle maintains constant velocity
after reaching its cruising speed. (a) For what time
interval is the bicycle ahead of the car? (b) By what maximum
distance does the bicycle lead the car?

Doing part a. 2. Relevant equations

3. The attempt at a solution
So my reasoning was that I need the position functions. I found them. I found the time at which the bike reaches its maximum velocity and selected that to be t=0. I adjusted both of the positions functions to this new setting.
Then I just went to solve for at what t they meet and my plan was just to add the (20/13) to that t and I would have the time at which the car overtakes the bike. However, I'm not successful in getting the right answer.
Where am I going wrong?

Did you account for velocity being in miles-per-hour and time being in seconds?
Note: You didn't need to reset the t=0 to when the bike hit top speed.

You reasoned that the bike is still ahead of the car at t=(20/13)s (did you check?)
Also that the car overtakes the bike while it is still accelerating, i.e. sometime before t=(50/9)s. (did you check?)

I tried messing with them and it only made everything go wrong. I have no idea why. I know that my t=20/13 is correct because of my solutions manual.
Yes I know that the bike is ahead until then because the bike accelerates faster, so until it stops accelerating it will always be ahead.

If the t for when the car overtook the bike would be greater than 50/9 then I'd know that the car was no longer accelerating when it took over the bike and I'd have to adjust for it. But that's not the case according to the solutions manual so no need to worry about that.

As for re-adjusting for t=0 I did that because the graph of the position of the motorcycle becomes linear at that point. The only other way that I know of graphing it would be piece wise but that seems like a lot of work for no apparent reason. The car's position function remains the same because it is still accelerating, now it only has an initial velocity which I accounted for already.

Well - to find the intersection of the linear part of the bikes displacement with the quadratic part of the car's displacement, you don't need to shift the origin. What is the equation of that line? (It only has to match the bikes displacement after t=20/13s - that it is different before then does not matter).

But it is not clear that you have taken the change in units into account ... i.e. you got the equations of motion by integrating the acceleration twice ...

So you start with a=9mph/s, you go to v(t)= 9t(mph), then to x(t)=9t^{2}/2 ... what are the units?
(eg. if this acceleration was held for 2s, did the bike travel 9(2)^{2}/2 = 18 what?)