Proof of Kinematics Equation: Eliminating Time from the Equation

[PurelyPhysical]

Homework Statement

Eliminate t from the equation (x-xi)=vi(t)+1/2(a)t^2 using the kinematic equation v=vi+at to get
v^2=vi^2+2a(x-xi)

The Attempt at a Solution

I wind up with (x-xi)=vi(v-vi/a) + 1/2(v^2-vi^2/a). If the first term on the right side didn't exist, I could see what the solution is, but I can't see why it just seems to disappear.


here is a video showing the exact step where I get lost. Again, the first term on the right just disappears. I can see why that would happen if the initial velocity is 0, but what if it isn't?

 

[haruspex]

You've marked several places in the video. Not sure which step is bothering you.
It might be more useful to post your own working.