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Find a+b+c

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,676
The real root of the equation \(\displaystyle 8x^3-3x^2-3x-1=0\) can be written in the form \(\displaystyle \frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}\), where \(\displaystyle a,\;b,\,c\) are positive integers.

Find \(\displaystyle a+b+c\).
 

mathworker

Active member
May 31, 2013
118
\(\displaystyle 8x^3-3x^2-3x-1=0\)
\(\displaystyle 9x^3=(x+1)^3\)
\(\displaystyle 9=(1+\frac{1}{x})^3\)
\(\displaystyle x=\frac{1}{\sqrt[3]{9}-1}\)
by factorizing,
\(\displaystyle a=81,b=9,c=8\)
\(\displaystyle a+b+c=98\)
 
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