For the equation x² – 5x + 6sgn x = 0 where sgn x denotes the signum function

$sig\left(c\right)=\left\{\begin{array}{l}-1 , x<0\\ 0, x=0\\ 1, x>0\end{array}\right.$

$case\left(ii\right) \begin{array}{l}\mathrm{case}\left(\mathrm{i}\right) \mathrm{x}>0\\ \Rightarrow {\mathrm{x}}^2-5\mathrm{x}+6=0\Rightarrow \mathrm{x}=2,3\\ x=0\\ \Rightarrow {\mathrm{x}}^2-5\mathrm{x}=0\Rightarrow \mathrm{x}=0\\ \mathrm{case}\left(\mathrm{iii}\right) \mathrm{x}<0\\ \Rightarrow {\mathrm{x}}^2-5\mathrm{x}-6=0\Rightarrow \mathrm{x}=-1\end{array}$