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May 14th, 2015, 08:37 AM  #1 
Newbie Joined: May 2015 From: India Posts: 14 Thanks: 0 Math Focus: Number Theory  Number of Solutions
Find the number of different real numbers x that satisfy (x^2+4x−2)^2 =(5x^2−1)^2

May 14th, 2015, 08:58 AM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902  Quote:
The first, x^2+ 4x 2= 5x^2 1 is the same as 4x^2 4x+ 1= 0 and x^2+ 4x= 1 5x^2 is the same 6x^2+ 4x 1= 0.  
May 14th, 2015, 09:23 AM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
This is asking for the number of distinct real roots of 24x^4  8x^3  22x^2 + 16x  3. You can factor this polynomial as $$(ax+b)^2(cx^2+dx+e)$$ and check whether the discriminant of the latter is positive, negative, or 0. 
May 14th, 2015, 09:33 AM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond 
Country Boy missed two cases.

May 16th, 2015, 04:38 PM  #5 
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  ?? Either both quadratics will be the same sign, or they will be the opposite sign. Country Boy's first resultant quadratic equation gives the repeated x = 1/2. His second resultant quadratic equation gives two distinct irrational values for x. Country Boy's two resultant quadratic equations will give the same number of solutions as the fourth degree equation posted above by CRGreathouse. 

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