HMMT November 2011 U32

Find all real numbers $x$ satisfying

$x^{9}+\frac{9}{8} x^{6}+\frac{27}{64} x^{3}-x+\frac{219}{512}=0$

Dataset text NuminaMath olympiads. Quoted with source.

Show the answer

x=\frac{1}{2}, \frac{-1 \pm \sqrt{13}}{4}

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