radical question again

*To*: mathgroup at smc.vnet.net*Subject*: [mg72064] radical question again*From*: "dimitris" <dimmechan at yahoo.com>*Date*: Sun, 10 Dec 2006 04:49:22 -0500 (EST)

Before proceeding to the post let me thank Andrzej and Daniel for their responses to my previous question appeared here http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_thread/thread/78d57749388d7384/e072ef4f6aa4f9c0?hl=en#e072ef4f6aa4f9c0 So consider again the following equation req = (2 - x)^2 - 4*Sqrt[1 - x]*Sqrt[1 - c*x] == 0; with 0<x<1 and 0<c<1. A quick solution is provided by Reduce[{req, 0 < x < 1, 0 < c < 1}, {x}, Reals] 0 < c < 1 && x == Root[-16 + 16*c + (24 - 16*c)*#1 - 8*#1^2 + #1^3 & , 1] Here comes my first question: Why the following setting returns the same output? Reduce[{req,0<x<1,0<c<1},{x},Reals,Cubics->True(*the default is False*)] 0 < c < 1 && x == Root[-16 + 16*c + (24 - 16*c)*#1 - 8*#1^2 + #1^3 & , 1] The following plots verify the obtained solution by Reduce f[c_] = x /. ToRules[Last[%]]; Plot[f[c], {c, 0, 1}] Plot[Chop[First[req] /. x -> f[c]], {c, 0, 1}] Let g[c_] = ToRadicals[f[c]] 8/3 - (8 - 48*c)/(6*(-17 + 45*c + 3*Sqrt[3]*Sqrt[11 - 62*c + 107*c^2 - 64*c^3])^(1/3)) + (2/3)*(-17 + 45*c + 3*Sqrt[3]*Sqrt[11 - 62*c + 107*c^2 - 64*c^3])^(1/3) with g[c] == x /. DeleteCases[Solve[req, x], {x -> 0}][[1]] True Now Block[{Message}, Plot[g[c], {c, 0, 1}, PlotPoints -> 100]] Show[GraphicsArray[Block[{$DisplayFunction = Identity}, (Plot[Chop[#1[g[c]]], {c, 0, 1}, Frame -> True, Axes -> False, PlotPoints -> 100] & ) /@ {Re, Im}]], ImageSize -> 500] The graphs can be justified due the choise of principal cube root by Mathematica. In the case of negative numbers, it turns out that it is not a real number. (see relevant descussion here: http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_thread/thread/403b579399a8b61/3d24f97466c328ed?hl=en#3d24f97466c328ed ) And here comes my second question: With the Root object we obtain a closed form root which is valid for 0<x,c<1. Converting it to radical we lost that and we must consider in more detail the range of c. Is it possible to get a radical expression which is equivalent to the Root object? If yes how? Thanks Dimitris