\(x^3-5x^2+5x-1=0\)
=>\(\left(x-1\right)\left(x^2+x+1\right)-5x\left(x-1\right)=0\)
=>\(\left(x-1\right)\left(x^2-4x+1\right)=0\)
=>\(\left[{}\begin{matrix}x=1\\x^2-4x+1=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\\left(x-2\right)^2=3\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}x=1\\x-2=\sqrt{3}\\x-2=-\sqrt{3}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\x=2+\sqrt{3}\\x=2-\sqrt{3}\end{matrix}\right.\)
\(S=\dfrac{1}{x_1^2}+\dfrac{1}{x_2^2}+\dfrac{1}{x_3^2}\)
\(=\dfrac{1}{1^2}+\dfrac{1}{\left(2-\sqrt{3}\right)^2}+\dfrac{1}{\left(2+\sqrt{3}\right)^2}\)
\(=1+\dfrac{1}{7-4\sqrt{3}}+\dfrac{1}{7+4\sqrt{3}}=1+7+4\sqrt{3}+7-4\sqrt{3}\)
=1+7+7
=15
