No. Using gcse physics terminology, momentum is a vector quantity (as its essetially velocity, so sign indicates direction) whereas relative
speed is a
scalar quantity so is
positive. The absolute value or modulus function is implied when you talk about
relative speeds being equal and is how resitution is usually stated
https://en.wikipedia.org/wiki/Coefficient_of_restitutionso
|u-v| = |U-V|where lower case is pre impact velociities and upper case is post impact velocities. Note that either u-v or U-V will be negative, but not both, as the objects must be approaching pre impact, but seperating post impact. so there is a sign change in the relative velocity in the collision. So your 20-12=8 and 10-#=8 must be wrong, as the first one is positive so the latter must be negative or 10-#=-8, so #=18.
So we express the relative speed relationship using the modulus or absolute value function around the relative velocities. Its an easy thing to overlook and using speeds (modulus / absolute value function) means that we dont care about whether the relative velocities are positive or negative or ... but we have to be a bit careful about applying it.
The relative speeds are equal (perfectly elastic) relationship is derived in
https://en.wikipedia.org/wiki/Elastic_collisionand its just a few lines from conserving KE and rearranging, dots, and dividing by momentum to get there. Its worth being clear that its really just the conservation of KE in another, simpler form. When you do it, you get something like
u-v = V-U
so the difference / relative velocity is flipped pre/post collision. Putting a modulus / absolute value function round each side means they represent relative speed as both are positive.