Question
Suppose a relativistic particle of mass $M$ and initial momentum $p_i$ is fired towards a stationary electron of mass $m_e$. Following the collision, the two particles move in the same direction with the electron having momentum $p_e$ and the particle now having momentum $p_f$. Show that the kinetic energy of the electron is given by $$ T = \dfrac{2 m_e c^6 p_i^2}{m_e^2 c^4 + M^2 c^4 + 2m_e c^2 \sqrt{p_i^2 c^2 + M^2 c^4}} $$
What I've tried before is writing down the conservation of momentum and energy, namely $p_i = p_e + p_f $ and $E_i = E_e + E_f$ along with the mass energy relation $ E^2 = p^2 c^2 + m^2 c^4 $ for each of the particles. In addition to this, I noted that the kinetic energy is given by $ E = T + mc^2 $ allowing us to derive that $ E^2 = T^2 + 2Tmc^2 + m^2 c^4 = p^2 c^2 + m^2 c^4 $, hence $ T^2 + 2Tmc^2 = p^2 c^2$ with appropriate subscripts for each particle. I'm not sure how to eliminate both $p_e$ and $p_f$ to get the final result into the desired form without getting lost in a sea of algebra. I've heard about four-vectors and their applications to these types of problems, although we haven't directly covered them in class so I don't think this is the desired approach.