Math CentralQuandaries & Queries


Question from Nazrul, a teacher:

If x^3+1/x^3=18*sqrt(3) then how can I evaluate the value of x+1/x?
Thank you for your help.


The most straightforward way, not needing any specialized algebraic tricks, is to substitute u = x3 and get

u + 1/u = 18 sqrt(3)

which is equivalent to the quadratic

u2 - 18 sqrt(3) + 1 = 0

which may be solved by standard means.

There are also many specialized tricks for problems of this sort. You can find out about these in books intended for math competition training. Herman, Kucera and Simsa' s book on "Equations and
Ineqalities" (CMS/Springer) would be a good place to start.

Good Hunting!



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