Relative motions, between inertial and non-inertial frames or bodies

I had no other way to put my problem to you guys thus I am taking an example….

Here,

aA = constant acceleration of person A

Ub = initial velocity of person B 

X = initial velocity of person A 

y = constant acceleration of the A   

Z = Initial distance between B and A

Now,I have seen people solving problems like this by assuming the frame of reference as "B".which is an non inertial frame of reference,and then using the equations of motion to solve the time taken by A to catch B.For example,the solution looks like….

My problem starts when,I start solving things from frame of reference of object A i.e an non-inertial frame of reference.The signs change and the final equation looks like-

Now,it is obvious that both the equations will not yield same results,so can you fix this thing i.e tell me where I am going wrong…….

By the way t is the time taken by A to catch B…..