ĐK: \(x\ge\dfrac{-2}{3}\)
\(9x^2+6x+1-4x^2+2.2x\sqrt{27x+18}-\left(27x+18\right)=0\)
\(\Leftrightarrow\left(3x+1\right)^2-\left(2x-\sqrt{27x+18}\right)^2=0\)
\(\Leftrightarrow\left(x+1+\sqrt{27x+18}\right)\left(5x+1-\sqrt{27x+18}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x+1+\sqrt{27x+18}=0\left(1\right)\\5x+1-\sqrt{27x+18}=0\left(2\right)\end{matrix}\right.\)
Xét (1): \(x+1+\sqrt{27x+18}=0\)
Do \(x\ge\dfrac{-2}{3}\Rightarrow x+1>0\Rightarrow VT>0\Rightarrow\) pt vô nghiệm
Xét (2): \(5x+1-\sqrt{27x+18}=0\) \(\Leftrightarrow5x+1=\sqrt{27x+18}\)
\(\Leftrightarrow\left\{{}\begin{matrix}5x+1\ge0\\\left(5x+1\right)^2=27x+18\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge\dfrac{-1}{5}\\25x^2-17x-17=0\end{matrix}\right.\)
\(\Rightarrow x=\dfrac{17+3\sqrt{221}}{50}\)