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Compton scattering

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I'm looking at instructor notes for one of the problems in my textbook, the link is here:
http://www.physics.smu.edu/~coan/hw_sols/5382/f02/5382_02_hw_sols_02.pdf

On page (3), where it says "Part B" P'Max occurs when...., can you please explain to me how they got the equations directly below till the end of the Page (3).

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What you do is you write down the equation for conservation of momentum. You define an x-direction, so that the electron moves in the positive x-direction and the backscattered photon moves in the negative x-direction. The momentum of a photon of wavelength lambda is h/(lambda). The backscattered photon has a ...

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