The sum of the squares of the roots of \(|x-2{|}^{2}+|x-2|-2=0\) and the squares of the roots of \({x}^{2}-2|x-3|-5=0\),…

\(|x−2{|}^{2}+|x−2|−2=0\)

\((|x−2|+2)(|x−2|−1)=0\)

\(|x−2|+2=0,\text{ }|x−2|−1=0\)

(not possible), \(|x−2|=1\)

\(x=2\pm 1\)

\(x=3,1\)

and \({x}^{2}−2(x−3)−5=0\)

\(\text{Case I:-  }x\text{ }⩾\text{ }3\)

\({x}^{2}−2x+6−5=0\)

\({(x−1)}^{2}=0\)

\(x=1\) (not possible)

\(\text{Case II:-  }x\text{ }<\text{ }3\)

\({x}^{2}+2x−6−5=0\)

\({x}^{2}+2x−11=0\)

Let \(\alpha ,\beta\) be the roots of this equation then

\({\alpha }^{2}+{\beta }^{2}={(\alpha +\beta )}^{2}−2\alpha \beta\)

\(={(−2)}^{2}−2(−11)=4+22=26\)

Sum of the squares of the roots \(={3}^{2}+{1}^{2}+26=36\)