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Penta-Quadratic (Posted on 2014-02-23) Difficulty: 3 of 5
Let x be a non-zero real number such that (x3+20x)1/5=(x5-20x)1/3. Find the product of all possible values of x.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 4.0000 (1 votes)

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Solution A long way | Comment 1 of 5
First the short way.  The apparent solutions from graphing the equations are x=0 and x=√5
So the product of the non-zero solutions is -5

I wanted to confirm this so I raised both sides to the 15th power, then simplified the result to a 22nd degree polynomial with all even terms.

Using the substitution y=x brings us to a 10th degree polynomial (too big to type out here)
I was able to use synthetic division to show y=5 is a zero of the polynomial so x=5 and so indeed x=√5

I was also able to find the some non-real solutions y=x=-4 so x= 2i and also an irrational y=x≈.0024990669
  Posted by Jer on 2014-02-23 21:35:56
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