The sum, of the squares of all the roots of the equation $x^2+|2 x-3|-4=0$, is

The sum, of the squares of all the roots of the equation $x^2+|2 x-3|-4=0$, is
  1. $3(3-\sqrt{2})$
  2. $6(3-\sqrt{2})$
  3. $6(2-\sqrt{2})$
  4. $3(2-\sqrt{2})$

Solution

$\begin{aligned} & \text { For } x \geq \frac{3}{2} \\ & x^2+2 x-3-4=0 \\ & x^2+2 x-7=0 \\ & x=\frac{-2 \pm \sqrt{4+28}}{2}=-1 \pm 2 \sqrt{2}\end{aligned}$
Only $2 \sqrt{2}-1$ is acceptable root
For $x < \frac{3}{2}$
$x^2-2 x+3-4=0$
$\begin{aligned}
& x^2-2 x-1=0 \\ & x=\frac{2 \pm \sqrt{4+4}}{2}=1 \pm \sqrt{2}
\end{aligned}$
Only $1-\sqrt{2}$ is acceptable
Sum of the square $=(1-\sqrt{2})^2+(2 \sqrt{2}-1)^2$
$=6(2-\sqrt{2})$ ~

Asked in: JEE Main 2025 (28 Jan Shift 1)

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