ĐKXĐ: \(x\ge\dfrac{-1}{4}\)
\(4x+1+\dfrac{2}{3}\sqrt{4x+1}+\dfrac{1}{9}-\left(3x\right)^2+2.\left(3x\right).\dfrac{11}{3}-\dfrac{121}{9}=0\)
\(\Leftrightarrow\left(\sqrt{4x+1}+\dfrac{1}{3}\right)^2-\left(3x-\dfrac{11}{3}\right)^2=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{4x+1}+\dfrac{1}{3}=3x-\dfrac{11}{3}\\\sqrt{4x+1}+\dfrac{1}{3}=\dfrac{11}{3}-3x\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{4x+1}=3x-4\left(1\right)\\\sqrt{4x+1}=\dfrac{10}{3}-3x\left(2\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow\left\{{}\begin{matrix}3x-4\ge0\\4x+1=\left(3x-4\right)^2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge\dfrac{4}{3}\\9x^2-28x+15=0\end{matrix}\right.\) \(\Rightarrow x=\dfrac{14+\sqrt{61}}{9}\)
\(\left(2\right)\Leftrightarrow\left\{{}\begin{matrix}\dfrac{10}{3}-3x\ge0\\4x+1=\left(\dfrac{10}{3}-3x\right)^2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\le\dfrac{10}{9}\\9x^2-24x+\dfrac{91}{9}=0\end{matrix}\right.\) \(\Rightarrow x=\dfrac{12-\sqrt{53}}{9}\)
