Correct option is: (3) 1
\(x|x-3|+3|x-2|+1=0\)
Case-I
\(x \leq 2\)
\(x(3-x)+3(2-x)+1=0\)
\(3 x-x^{2}+6-3 x+1=0\)
\(x^{2}=7\)
\(x= \pm \sqrt{7} \quad \quad x=-\sqrt{7}\)
Case-II
\(2<x \leq 3\)
\(x(3-x)+3(x-2)+1=0\)
\(3 x-x^{2}+3 x-6+1=0\)
\(x^{2}-6 x+5=0\)
\((x-5)(x-1)=0\)
\(\Rightarrow x=5,1\) (no solution)
Case-III
\(x \geq 3\)
\(x(x-3)+3(x-2)+1=0\)
\(x^{2}-3 x+3 x-6+1=0\)
\(x^{2}=5\)
\(\Rightarrow x= \pm \sqrt{5}\) (no solution)
Only 1 solution i.e., \(-\sqrt{7}\)
