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Two Power Equation Poser (Posted on 2015-12-05) Difficulty: 3 of 5
Determine all possible real numbers x that satisfy this equation:

2(2x- 1)x2 + (2x^2- 2)x = 2x+1 - 2

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 1 of 1
First by inspection there are solutions x={-1,0,1}

Second rewrite the equation a bit
2(2x- 1)x2 + (2x^2- 2)x = 2x+1 - 2
2(2x- 1)x2 + (2x^2- 2)x = 2(2x - 1)
2(2x- 1)x2 - 2(2x - 1)= -(2x^2- 2)x
2(2x- 1)(x2 -1) = -(2x^2- 2)x
2(2x- 1)/(2x^2- 2) = -x/(x2 -1)

Third compare each by intervals
(2x- 1) is - on (-inf,-0) and + on (0,inf)
(2x^2- 2) is - on (-1,1) and - on (-inf,-1), (1,inf)

-x is + on (-inf,0) and - on (0,inf)
(x2 -1) is - on (-1,1) and + on (-inf,-1), (1,inf)

Finally, since these quotients are each + where the other is - they can never be equal.  So there are no other solutions.


  Posted by Jer on 2015-12-06 17:29:16
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