\(\left(x+3\sqrt{x}+2\right)\left(x+9\sqrt{x}+18\right)=168x\left(x\ge0\right)\\ \Leftrightarrow\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)\left(\sqrt{x}+3\right)\left(\sqrt{x}+6\right)=168x\\ \Leftrightarrow\left(x+7\sqrt{x}+6\right)\left(x+5\sqrt{x}+6\right)=168x\\ \Leftrightarrow\left(\sqrt{x}+\dfrac{6}{\sqrt{x}}+7\right)\left(\sqrt{x}+\dfrac{6}{\sqrt{x}}+5\right)=168\)
Đặt \(\sqrt{x}+\dfrac{6}{\sqrt{x}}=t\), pt trở thành
\(\left(t+7\right)\left(t+5\right)=168\\ \Leftrightarrow t^2+12t+35=168\\ \Leftrightarrow t^2+12t-133=0\\ \Leftrightarrow\left[{}\begin{matrix}t=7\\t=-19\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\sqrt{x}+\dfrac{6}{\sqrt{x}}=7\\\sqrt{x}+\dfrac{6}{\sqrt{x}}=-19\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x-7\sqrt{x}+6=0\\x+19\sqrt{x}+6=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x}=1\\\sqrt{x}=6\\x+19\sqrt{x}+6=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\left(tm\right)\\x=36\left(tm\right)\\x+19\sqrt{x}+6=0\left(\text{*}\right)\end{matrix}\right.\)
Đặt \(\sqrt{x}=a\), thay vào \(\left(\text{*}\right)\), ta được:
\(a^2+19a+6=0\\ \Delta=361-4\cdot6=337\\ \Leftrightarrow\left[{}\begin{matrix}a=\dfrac{\sqrt{337}-19}{2}\\a=\dfrac{\sqrt{337}+19}{2}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{356-38\sqrt{337}}{4}\\x=\dfrac{356+38\sqrt{337}}{4}\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{178-19\sqrt{337}}{2}\left(ktm\right)\\x=\dfrac{178+19\sqrt{337}}{2}\left(tm\right)\end{matrix}\right.\)
Vậy \(S=\left\{1;36;\dfrac{178+19\sqrt{337}}{2}\right\}\)
