       Re: Eliminate Complex Roots

• To: mathgroup at smc.vnet.net
• Subject: [mg64361] Re: [mg64348] Eliminate Complex Roots
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Tue, 14 Feb 2006 01:31:36 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Reduce[a^3+10*a^2-15*a+b==0,a,Reals]

(b < (10/27)*(-335 - 29*Sqrt) && a == Root[#1^3 + 10*#1^2 - 15*#1
+ b & , 1]) ||
(b == (10/27)*(-335 - 29*Sqrt) &&
(a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 1] ||
a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 3])) ||
((10/27)*(-335 - 29*Sqrt) < b < (10/27)*(-335 + 29*Sqrt) &&
(a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 1] ||
a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 2] ||
a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 3])) ||
(b == (10/27)*(-335 + 29*Sqrt) &&
(a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 1] ||
a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 2])) ||
(b > (10/27)*(-335 + 29*Sqrt) && a == Root[#1^3 + 10*#1^2 - 15*#1
+ b & , 1])

Reduce[{a^3+10*a^2-15*a+b\[Equal]0,b==(10/27)*(-335-29*Sqrt)},a]

b == (10/27)*(-335 - 29*Sqrt) && (a == (1/3)*(-10 - Sqrt) ||
a == (2/3)*(-5 + Sqrt))

%//N

b == -253.4097195499913 && (a == -7.347198192930765 || a ==
4.6943963858615305)

Plot[Root[#1^3+10*#1^2-15*#1+b&,1],{b,-500,25}];

Bob Hanlon

>
> From: bghiggins at ucdavis.edu
To: mathgroup at smc.vnet.net
> Subject: [mg64361] [mg64348] Eliminate Complex Roots
>
> Hi All,
>
> Suppose I have the following equation and I want to plot the real roots
> using Plot as a function of the parameter b
>
> a^3 + 10*a^2 - 15*a + b == 0
>
> For the problem I am interested I have a set of algebraic equations of
> at least order 3, but this is simpler enough to explain what the issue
> is.
>
> Now if I want just the Real root I can do the following:
>
>
> Sol2[b_] := DeleteCases[Solve[a^3 + 10.*a^2 - 15*a + b ==
>      0, a], {___, x_ -> Complex[y_, z_], ___}]
>
> and evaluating
>
> Sol2
>
> {{a -> -11.46104034060055}}
>
> Gives the desired result. So far so good. Now suppose I want to plot
> the solutions asa function of b. My immediate thought was to try
>
> Plot[Evaluate[a /. Sol2[b]], {b, -10, 30}, PlotStyle -> {Red, Blue,
> Magenta}]
>
> This does give the desired result, but gives error messages when it
> tries to plot a complex quantity. That is the DeleteCases does not seem
> to work within Plot, even though I wrapped everything with evaluate.
>
> Any suggestions?
>
> Thanks much
>
> Brian
>
>

```

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