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Tuesday, December 26, 2017

2017/029) Find the exact value for the real root of the equation x^3+3x-2=0

Say f(x)=x3+3x-2=0
f(x)  changes sign once so there is one positive root.
f(-x)=-x^3-3x-2 changes sign zero times so no -ve root
let x=2sinh\, t
so x^3+3x=2=>8\sinh\,3t+6sinh\,t=2
or 4sinh\,3t+3sinh\,t=1
or sinh\,3t=1 or 3t=sinh^{-1}1
hence x=2sinh(\frac{sinh^{-1}}{3})x
this is the only real root

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