1. **State the problem:** Solve the system of equations: $$x + y = 2$$ $$xy = 4$$ 2. **Use the quadratic equation approach:** From the first equation, we know the sum of $x$ and $y$ is 2. 3. **Recall the relationship:** If $x$ and $y$ are roots of a quadratic equation, then the equation can be written as: $$t^2 - (x+y)t + xy = 0$$ 4. **Substitute the known values:** $$t^2 - 2t + 4 = 0$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 1 \times 4 = 4 - 16 = -12$$ 6. **Interpret the discriminant:** Since $\Delta < 0$, the roots are complex (no real solutions). 7. **Find the roots using the quadratic formula:** $$t = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{2 \pm \sqrt{-12}}{2} = 1 \pm i\sqrt{3}$$ 8. **Final answer:** $$x = 1 + i\sqrt{3}, \quad y = 1 - i\sqrt{3}$$ or vice versa, where $i$ is the imaginary unit.