The sum, of the squares of all the roots of the equation , is [2025]

Question

The sum, of the squares of all the roots of the equation $x^{2} + | 2 x - 3 | - 4 = 0$ , is [2025]

Smart Achievers · Accepted Answer

(1)

Given, $x^{2} + | 2 x - 3 | - 4 = 0$

Case I: $x^{2} + (2 x - 3) - 4 = 0 when x \geq \frac{3}{2}$

$\Rightarrow x^{2} + 2 x - 7 = 0 \Rightarrow x = 2 \sqrt{2} - 1$ $[∵ x \geq \frac{3}{2}]$

Case II : $x^{2} - (2 x - 3) - 4 = 0, x < \frac{3}{2}$

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

$\Rightarrow x = \frac{2 \pm 2 \sqrt{2}}{2} = 1 \pm \sqrt{2} \Rightarrow x = 1 - \sqrt{2}$ $[∵ x < \frac{3}{2}]$

$∴$ Required sum = ${(2 \sqrt{2} - 1)}^{2} + {(1 - \sqrt{2})}^{2}$

$= 12 - 6 \sqrt{2} = 6 (2 - \sqrt{2})$ .

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