1. **State the problem:** Solve the quadratic equation $2x^2 - x + 2 = 0$. 2. **Formula used:** The quadratic formula is used to solve equations of the form $ax^2 + bx + c = 0$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $a = 2$, $b = -1$, and $c = 2$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-1)^2 - 4 \times 2 \times 2 = 1 - 16 = -15$$ 5. **Interpret the discriminant:** Since $\Delta < 0$, there are no real roots; the solutions are complex. 6. **Find the complex roots:** $$x = \frac{-(-1) \pm \sqrt{-15}}{2 \times 2} = \frac{1 \pm \sqrt{15}i}{4}$$ 7. **Final answer:** $$x = \frac{1}{4} + \frac{\sqrt{15}}{4}i \quad \text{or} \quad x = \frac{1}{4} - \frac{\sqrt{15}}{4}i$$ These are the two complex solutions to the quadratic equation.