Ta có: \(6x\left(3x+5\right)-2x\left(3x-2\right)+\left(17-x\right)\left(x-1\right)+x\left(x-18\right)=0\)
\(\Leftrightarrow18x^2+30x-6x^2+4x+17x-17-x^2+x+x^2-18x=0\)
\(\Leftrightarrow12x^2-34x-17=0\)
\(\Leftrightarrow12\left(x^2-\frac{34}{12}x-\frac{17}{12}\right)=0\)
\(\Leftrightarrow x^2-2\cdot x\cdot\frac{17}{12}+\frac{289}{144}-\frac{493}{144}=0\)
\(\Leftrightarrow\left(x-\frac{17}{12}\right)^2=\frac{493}{144}\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\frac{17}{12}=\frac{\sqrt{493}}{12}\\x-\frac{17}{12}=-\frac{\sqrt{493}}{12}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\frac{17+\sqrt{493}}{12}\\x=\frac{17-\sqrt{493}}{12}\end{matrix}\right.\)
Vậy: \(S=\left\{\frac{17+\sqrt{493}}{12};\frac{17-\sqrt{493}}{12}\right\}\)