1. **State the problem:** Solve the quadratic equation $2x^2 - x + 5 = 0$. 2. **Formula used:** The quadratic formula for $ax^2 + bx + c = 0$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-1$, and $c=5$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-1)^2 - 4 \times 2 \times 5 = 1 - 40 = -39$$ Since $\Delta < 0$, the equation has no real roots but two complex roots. 4. **Find the roots:** $$x = \frac{-(-1) \pm \sqrt{-39}}{2 \times 2} = \frac{1 \pm \sqrt{-39}}{4} = \frac{1 \pm i\sqrt{39}}{4}$$ 5. **Final answer:** $$x = \frac{1}{4} + \frac{i\sqrt{39}}{4} \quad \text{or} \quad x = \frac{1}{4} - \frac{i\sqrt{39}}{4}$$ These are the two complex solutions to the quadratic equation.