problem 246 mathcounts

Mathematics
Tutor: None Selected Time limit: 1 Day

If f(x)=x(x-1)(x+1), what is the product of the nonzero real numbers x such that f(x)=x

Oct 12th, 2014

1. f(1/x) + 2f(x) = x for all nonzero real x... therefore, the same should hold true if you substitute 1/x for x: 

2. f(x) + 2f(1/x) = 1/x 

Multiply the original equation through by 2: 

3. 2f(1/x) + 4f(x) = 2x 

Now substract equation #2 from equation #3, and solve for f(x): 

( 2f(1/x) + 4f(x) ) - ( f(x) + 2f(1/x) ) = 2x - 1/x 
2f(1/x) - 2f(1/x) + 4f(x) - f(x) = 2x^2/x - 1/x 
3f(x) = (2x^2 - 1)/x 
f(x) = (2x^2 - 1)/3x

Oct 12th, 2014

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