This is good advice. Here are some additional details for the OP:
1) Dig more space around the pipe so you have room to straighten the pipe once the root is out of the way.
2) Cut the root out so that it's no longer bending the pipe at all. Personally I'd use a handsaw over a chainsaw or sawzall to minimize the risk of accidentally cutting the pipe (PVC is soft).
3) Cut the PVC pipe back beyond the weird bulge one one side, and a few inches past it on the other.
4) Buy a matching diameter PVC pipe that's long enough to fill the gap several times over. Also buy at least four PVC couplers of the appropriate diameter (they're smooth on the inside and slide over the long pipe segment you just bought). Also buy some PVC cement. It's used to bond the couplers and pipe segments together.
5) Cut the new pipe segment and slide it and the couplers onto the existing pipes to make sure everything lines up straight and easily like so: ==OldPipe==Coupler==NewPipe==Coupler==OldPipe==
6) Realize you cut something wrong or gouged a coupler or somesuch. Upvote me for telling you to spend the extra $5 on spare pipe and couplers so you can redo it right away rather than drive back to the hardware store.
7) Put on a painting mask. PVC Cement is highly...pungent...and will give you a headache.
8) Apply the PVC cement under the couplers and slide them over the old pipe ends and new segment until the cement dries (this only takes a few minutes). Give it a tug to verify that it's now as solid as one continuous pipe thanks to the PVC cement.
9) Turn the water on. If it holds pressure, great. If it leaks, cut it off, try again, use more PVC cement this time, and thank me for making you spend the extra $5 on more couplers to save you an extra trip to the hardware store if you mess it up.
Fortunately PVC is pretty forgiving, particularly outdoors where "cut more off and try again" is easy.
This is the advice I wish I was able to give :) Listen to this man.
And I love the graphic!!! (==OldPipe==Coupler==NewPipe==Coupler==OldPipe==) this is exactly the kind of visual I'd like to have...simple, but effective.
My nigger, couplings are NOT smooth on the inside, there is a ridge in the center. A Dremel tool or round hand file can smooth the ridge down so the coupling can be slid onto the pipe, and then slid over the break and glued.
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