\(\sqrt{x+1+\sqrt{x+\dfrac{3}{4}}}+x=\dfrac{1}{2}\)
\(\Leftrightarrow\sqrt{x+1+\dfrac{1}{2}\sqrt{4x+3}}+x=\dfrac{1}{2}\)
\(\Leftrightarrow\sqrt{\dfrac{1}{4}\left(4x+3\right)+2.\dfrac{1}{2}.\dfrac{1}{2}\sqrt{4x+3}+\dfrac{1}{4}}+x=\dfrac{1}{2}\)
\(\Leftrightarrow\sqrt{\left(\dfrac{1}{2}\sqrt{4x+3}+\dfrac{1}{2}\right)^2}+x=\dfrac{1}{2}\)
\(\Leftrightarrow\dfrac{1}{2}\sqrt{4x+3}+\dfrac{1}{2}+x=\dfrac{1}{2}\)
\(\Leftrightarrow\sqrt{4x+3}=-2x\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\le0\\4x+3=4x^2\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x\le0\\\left(2x-3\right)\left(2x+1\right)=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x\le0\\\left[{}\begin{matrix}x=\dfrac{3}{2}\\x=-\dfrac{1}{2}\end{matrix}\right.\end{matrix}\right.\)
\(\Rightarrow x=-\dfrac{1}{2}\)
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