Two cars $$P$$ and $$Q$$ start from a point at the same time in a straight line and their positions are represented by $${X_P}\left( t \right) = at + b{t^2}$$    and $${X_Q}\left( t \right) = ft - {t^2}.$$    At what time do the cars have the same velocity?

Solution :
Velocity of each car is given by
$$\eqalign{ & {V_P} = \frac{{d{x_P}\left( t \right)}}{{dt}} = a + 2bt \cr & {\text{and}}\,{V_Q} = \frac{{d{x_Q}\left( t \right)}}{{dt}} = f - 2t \cr} $$
It is given that $${V_P} = {V_Q}$$
$$\eqalign{ & \Rightarrow a + 2bt = f - 2t \cr & \Rightarrow t = \frac{{f - a}}{{2\left( {b + 1} \right)}} \cr} $$