Question 8

Given $|x^2 - 2x + 3| = 11$.

1. Analyze the Quadratic:

   $$x^2 - 2x + 3 = (x - 1)^2 + 2.$$

   For all real $x$, $(x - 1)^2 \ge 0$, which implies $x^2 - 2x + 3 \ge 2 > 0$.
   Because the term inside the absolute value is strictly positive, the equation simplifies to:

   $$x^2 - 2x + 3 = 11 \implies x^2 - 2x - 8 = 0.$$

2. Solve for $x$:

   $$(x - 4)(x + 2) = 0 \implies x = 4 \quad \text{or} \quad x = -2.$$

3. Evaluate Target Expression $|-x^3 + x^2 - x|$:

   • Case 1 ($x = 4$):
     $$-4^3 + 4^2 - 4 = -64 + 16 - 4 = -52.$$
     $$|-52| = 52.$$
   • Case 2 ($x = -2$):
     $$-(-2)^3 + (-2)^2 - (-2) = -(-8) + 4 + 2 = 8 + 4 + 2 = 14.$$
     $$|14| = 14.$$

4. Thus, the possible values are $14$ and $52$ (option D).