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Descartes roots (Posted on 2019-03-17) Difficulty: 3 of 5
Find the sum of all real numbers x such that
x = (3 - (29x - 1)1/2013) / 29

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Solution Comment 1 of 1
I started by rearranging the equation into (29x-3)^2013 + 29x - 1 = 0.  

Then I substituted y=29x-3 to get f(y)=y^2013+y+2=0.  There is one obvious real solution y=-1.

The first derivative f'(y)=2013*y^2012+1 is always greater than zero which means f(y) is continuously increasing and then must have only the one real root y=-1.

Then y=-1 implies x=2/29.

  Posted by Brian Smith on 2019-03-17 13:14:25
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