JEE Main · 2025 · 08 Apr 2025 · Shift 2 · Quadratic Equations and Inequalities

Read the first equation as $|x-2{|}^2+|x-2|-2=0$. Setting $t=|x-2|\ge 0$, $t^2+t-2=0$ gives $t=1$, so $|x-2|=1$ and $x=1$ or $x=3$; the sum of the squares of these roots is $1+9=10$. For $x^2-2|x-3|-5=0$: if $x\ge 3$ then $x^2-2x+1=0$ forces $x=1$, which is rejected; if $x<3$ then $x^2+2x-11=0$, whose two roots both satisfy $x<3$, with sum $-2$ and product $-11$, so the sum of their squares is $(-2{)}^2-2(-11)=26$. Adding the two parts gives $10+26=36$.