If \(x\) is real and \(\mid x^2-2x+3 \mid = 11\), then possible values of…
2014
If \(x\) is real and \(\mid x^2-2x+3 \mid = 11\), then possible values of \(\mid -x^3+x^2-x\mid\) include
- A.
2, 4
- B.
2, 14
- C.
4, 52
- D.
14, 52
Attempted by 29 students.
Show answer & explanation
Correct answer: D
Key idea: solve the absolute-value equation to find real x, then evaluate the expression for those x.
Solve |x^2 - 2x + 3| = 11.
Case 1: x^2 - 2x + 3 = 11 → x^2 - 2x - 8 = 0 → (x - 4)(x + 2) = 0, so x = 4 or x = -2.
Case 2: x^2 - 2x + 3 = -11 → x^2 - 2x + 14 = 0, which has negative discriminant, so no real roots.
Evaluate -x^3 + x^2 - x at the found real roots:
For x = 4: -x^3 + x^2 - x = -64 + 16 - 4 = -52, so the absolute value is 52.
For x = -2: -x^3 + x^2 - x = 8 + 4 + 2 = 14, so the absolute value is 14.
Conclusion: the possible values of | -x^3 + x^2 - x | are 14 and 52.