May 14th, 2012, 11:10 PM  #1 
Member Joined: Apr 2012 Posts: 72 Thanks: 3  find x
?( (1?(1x^2))/2 ) + ?( (1+?(1x^2))/2 ) = ?(1+x)

May 15th, 2012, 12:09 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: find x
We are given: We see we require in order for the left side to represent real values. Multiply through by : Square both sides since they are nonnegative: Case 1: Hence, we find: Case 2: No solution for this case, thus we are left with: 
May 15th, 2012, 12:35 AM  #3 
Member Joined: Apr 2012 Posts: 72 Thanks: 3  Re: find x
Thanks a lot!

May 15th, 2012, 06:13 AM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: find x
A sorta "short cut to typing!" I'd use here: since x^2  1 = (x+1)(x1): let a = x+1 and b = x1; so, after multiplication by SQRT(2), you have: SQRT[1  SQRT(ab)] + SQRT[1 + SQRT(ab)] = SQRT(2a) Easier to do the squaring both sides et al... I'm lazy 
May 15th, 2012, 09:55 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,942 Thanks: 2210 
Assuming x ? 1, let x = sin(2A), where 2A ? ?/2, and recall that cos(2A) = 2cos²(A)  1 = 1  2sin²(A). The equation becomes sin(A) + cos(A) = ?(1 + sin(2A)). Squaring gives 1 + sin(2A) = 1 + sin(2A), i.e. x = x, so 0 ? x ? 1. 

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