homework and exercises - Question regarding elastic collisions and the conservation of momentum / energy

I've been working on this question for a few hours and I cannot quite convince myself that my answer is correct. I've read the relevant section in my textbook, but I guess the lightbulb just isn't clicking on. The question (which I'll post the full text of below), asks for final velocity of a puck in three body collision in which two of the bodies are initially at rest. I was thinking that, since I only need final velocity, I could just use the conservation of linear momentum in the relevant plane (since this equation has a term for v) and solve it out for v. Other solution I've seen use both linear momentum AND conservation of energy equations. Which way is correct? See picture for question and attempted solution. Any help is greatly appreciated.

Question: Three identical round pucks A, B, and C are placed on frictionless surface as shown. The puck A is shot with initial velocity V0 along the symmetry line. The pucks B and C are initially touching each other. The pucks undergo elastic collision. This question asks that I find the final velocity of Puck A.

EDIT: My attempt is included in the image below. The problem I'm currently having is that I'm not sure if I can treat Pucks A & B as a single object for the purposes of this solution.