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an equation containg radicals

• To: mathgroup at smc.vnet.net
• Subject: [mg69648] an equation containg radicals
• From: dimmechan at yahoo.com
• Date: Tue, 19 Sep 2006 05:45:07 -0400 (EDT)

Hello to all.

In a crack problem appeared the following function.

K[p_] := 1 - 4*(1 - v)*λ^2*p^2*(1 - Sqrt[e^2 - p^2]/Sqrt[a^2 - p^2])
a = 1/λ;

Here are some typical values for the involving constants

consts = {e -> 1/1000, v -> 3/10, λ -> 10^(-5)};

Here is the solution obtained with Solve

sols = FullSimplify[Solve[eq = K[p] == 0, p]]
{{p -> (-Sqrt[2])*Sqrt[-(1/(λ^2*(-9 + 8*v + Sqrt[-15 + 16*v +
64*e^2*(-1 + v)^2*λ^2])))]},
  {p -> Sqrt[2]*Sqrt[-(1/(λ^2*(-9 + 8*v + Sqrt[-15 + 16*v + 64*e^2*(-1
+ v)^2*λ^2])))]},
  {p -> (-Sqrt[2])*Sqrt[1/(λ^2*(9 - 8*v + Sqrt[-15 + 16*v + 64*e^2*(-1
+ v)^2*λ^2]))]},
  {p -> Sqrt[2]*Sqrt[1/(λ^2*(9 - 8*v + Sqrt[-15 + 16*v + 64*e^2*(-1 +
v)^2*λ^2]))]}}

What I need now is to see which roots (or if all roots) are extreneous
(i.e. they do not satisfy the intial equation K[p]=0).
This is a difficult task for Mathematica.

TimeConstrained[FullSimplify[eq /. sols], 300, "Failed"]
"Failed"

However replacing the values for the constants it is verified that all
solutions (for this typical values of the constants) are extreneous.

eq /. sols /. consts
{False, False, False, False}

This can be verified also by the following command

Solve[eq /. consts, p]
{}

My question is if it a way to "help" a bit Mathematica in the above
verification with the symbolic parameters.

Thanks in advance for any help.

• Follow-Ups:
  • Re: an equation containg radicals
    • From: Daniel Lichtblau <danl@wolfram.com>