Math help

[Surgenator]

I'm going to use laTex syntax for this.

V_{b}(t) = v_b
V_{a}(t) = v_a + a*t

D_{b}(t) = integral (V_{b} dt) = v_b*t
D_{a}(t) = integral (V_{a} dt) = v_a*t + a*t*t/2
\Delta D(t) = D_{b}(t) - D_{a}(t) = (v_b - v_a)*t + a*t*t/2

Assuming 'a' is negative so that car A can come to stop. Car A will come to a stop when V_{a}(t) = 0.
V_{a}(t) = 0 = v_a + a*t_{stop}
after some algebra
t_{stop} = -v_a/a
I'm going to absorbed the negative into 'a'

Plug in the t_{stop} value into the \Delta D equation
\Delta D(t) = (v_b - v_a)*(v_a/a) + a*(v_a/a)*(v_a/a)/2
After a little bit of algebra
\Delta D(t) = (v_b*v_a - v_a*v_a)/a + (v_a*v_a)/(2*a)
\Delta D(t) = 2*(v_b*v_a - v_a*v_a)/(2*a) + (v_a*v_a)/(2*a)
\Delta D(t) = (2*v_b*v_a - 2*v_a*v_a + v_a*v_a)/(2*a)
\Delta D(t) = (2*v_b*v_a - v_a*v_a)/(2*a)
There is your answer