The sum of the squares of all the roots of the equation $x^2 + |2x - 3| - 4 = 0$ is
Step-by-Step Solution
Key Concept: Split into two cases based on the sign of $2x-3$ and solve each resulting quadratic, retaining only roots consistent with the case condition.
**Case 1** ($x \geq 3/2$): $|2x-3| = 2x-3$.
$x^2 + 2x - 7 = 0 \Rightarrow x = -1 + 2\sqrt{2}\approx1.83$ (valid; $\geq 3/2$). The other root $-1-2\sqrt{2}<0$ is rejected.
**Case 2** ($x < 3/2$): $|2x-3| = 3-2x$.
$x^2 - 2x - 1 = 0 \Rightarrow x = 1-\sqrt{2}\approx-0.41$ (valid; $<3/2$). The other root $1+\sqrt{2}>3/2$ is rejected.
**Sum of squares:**
$(1-\sqrt{2})^2 + (2\sqrt{2}-1)^2 = (3-2\sqrt{2}) + (9-4\sqrt{2}) = 12-6\sqrt{2} = 6(2-\sqrt{2})$.
Correct Answer: 3