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I can do this simple root equation by hand rather quickly for x=3/2. but Sage just throws the equality back at me.
Of course I don't need it for simple algebra, but I'm wondering what makes it choke, and what that means for more complex solutions.

var('x')
solve((x-1)^.5-(2*x-3)^.5-(3*x-4)^.5==0,x)

ans:
[sqrt(x - 1) == sqrt(3*x - 4) + sqrt(2*x - 3)]