JEE Main 2025 — 8 April, Evening Shift — Question 35

$|x-2|^{2}+|x-2|-2=0$

$\Rightarrow(|x-2|+2)(|x-2|-1)=0$

$\Rightarrow|x-2|=1$

$\Rightarrow x=3,1$

$x^{2}-2|x-3|-5=0$

$x \geq 3$

$x^{2}-2(x-3)-5=0$

$\Rightarrow x^{2}-2 x+1=0$

$\Rightarrow(x-1)^{2}=0$

$\Rightarrow x=1 \quad$ (rejected)

$x<3$

$x^{2}+2(x-3)-5=05$

$\Rightarrow x^{2}+2 x-11=0$

$$x=-1 \pm 2 \sqrt{3}$$

Sum of square of roots

$=3^{2}+1^{2}+(-1+2 \sqrt{3})^{2}+(-1-2 \sqrt{3})^{2}$

$=10+(1+12) 2$

$=36$