\(17-x=\frac{60}{x}\Leftrightarrow17x-x^2=60\Leftrightarrow17x-x^2-60=0\)
\(\Leftrightarrow-x^2+12x+5x-60=0\Leftrightarrow-x\left(x-12\right)+5\left(x-12\right)=0\)
\(\Leftrightarrow\left(5-x\right)\left(x-12\right)=0\Leftrightarrow\orbr{\begin{cases}5-x=0\\x-12=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=5\\x=12\end{cases}}\)
\(17-x=60:x\)
\(\frac{17-x}{1}=\frac{60}{x}\)
\(x\left(17-x\right)=60.1\)
\(17x-x^2=60\)
\(17x-x^2-60=0\)
\(12x+5x-x^2-60=0\)
\(\left(12x-x^2\right)+\left(5x-60\right)=0\)
\(x\left(12-x\right)+5\left(x-12\right)=0\)
\(x\left(12-x\right)-5\left(12-x\right)=0\)
\(\left(x-5\right)\left(12-x\right)=0\)
\(\Rightarrow\hept{\begin{cases}x-5=0\\12-x=0\end{cases}\Rightarrow\hept{\begin{cases}x=5\\x=12\end{cases}}}\)