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FullSimplify and negative radicands

• To: mathgroup at smc.vnet.net
• Subject: [mg110199] FullSimplify and negative radicands
• From: Torsten Schoenfeld <kaffeetisch at gmx.de>
• Date: Mon, 7 Jun 2010 08:23:04 -0400 (EDT)

Why does FullSimplify think this equation is true even though x = -3 is 
clearly a counterexample?

In[1]:= equation = Sqrt[x^2 - 1] == Sqrt[x + 1] Sqrt[x - 1]
Out[1]= Sqrt[-1 + x^2] == Sqrt[-1 + x] Sqrt[1 + x]

In[2]:= $Assumptions
Out[2]= True

In[3]:= equation // FullSimplify
Out[3]= True

In[4]:= equation /. x -> -3
Out[4]= False

(This is with Mathematica 7.0.1.0.)