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MathGroup Archive 2006

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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


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