Solve the absolute value equation or indicate that the equation has no solution.
-$$\left| 2 x ^ { 2 } - x - 1 \right| = 3$$
A) $\left\{ \frac { 1 - \sqrt { 33 } } { 4 } , - \frac { 1 + \sqrt { 33 } } { 4 } \right\}$
B) $\left\{ - \frac { 1 - \sqrt { 33 } } { 4 } , - \frac { 1 + \sqrt { 33 } } { 4 } \right\}$
C) $\left\{ \frac { 1 - \sqrt { 33 } } { 4 } , \frac { 1 + \sqrt { 33 } } { 4 } \right\}$
D) $\varnothing$

Question

Quizplus · Accepted Answer

To solve the equation $\left| 2x^2 - x - 1 \right| = 3$, we consider two cases due to the absolute value:1. $2x^2 - x - 1 = 3$2. $2x^2 - x - 1 = -3$For the first case, simplifying gives $2x^2 - x - 4 = 0$. Using the quadratic formula, $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-4)}}{2(2)}$, which simplifies to $x = \frac{1 \pm \sqrt{33}}{4}$.For the second case, simplifying gives $2x^2 - x + 2 = 0$. However, the discriminant of this quadratic equation is negative ($b^2 - 4ac = (-1)^2 - 4(2)(2) = -15$), indicating no real solutions.Therefore, the solutions are $x = \frac{1 - \sqrt{33}}{4}$ and $x = \frac{1 + \sqrt{33}}{4}$, which corresponds to choice C.

Solve the Absolute Value Equation or Indicate That the Equation ∣2x2−x−1∣=3\left| 2 x ^ { 2 } - x - 1 \right| = 3​2x2−x−1​=3

Solve the Absolute Value Equation or Indicate That the Equation $\left| 2 x ^ { 2 } - x - 1 \right| = 3$