\(x^2+x+\frac{1}{2}=\sqrt{2x^3-x^2+x+1}\)
\(\Leftrightarrow\left(x^2+x+\frac{1}{2}\right)^2=2x^3-x^2+x+1\)
\(\Leftrightarrow x^4+x^2+\frac{1}{4}+2x^3+x^2+x=2x^3-x^2+x+1\)
\(\Leftrightarrow x^4+2x^3+2x^2+x+\frac{1}{4}=2x^3-x^2+x+1\)
\(\Leftrightarrow x^4+3x^2-\frac{3}{4}=0\)
\(\Leftrightarrow\left(x^2+\frac{3}{2}\right)^2-3=0\)
\(\Leftrightarrow\left(x^2+\frac{3}{2}\right)^2=3\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2+\frac{3}{2}=\sqrt{3}\\x^2+\frac{3}{2}=-\sqrt{3}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2=\sqrt{3}-1,5\\x^2=-\sqrt{3}-\frac{3}{2}\left(VL\right)\end{matrix}\right.\)
\(\Leftrightarrow x=\pm\sqrt{\sqrt{3}-1,5}\)
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