1. The problem is to simplify the expression $$\frac{2 \pm \sqrt{(-2)^2 + 4(1)(1)}}{2}$$. 2. First, calculate the value inside the square root (the discriminant): $$(-2)^2 + 4(1)(1) = 4 + 4 = 8$$. 3. Substitute back into the expression: $$\frac{2 \pm \sqrt{8}}{2}$$. 4. Simplify the square root: $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$. 5. Now the expression is $$\frac{2 \pm 2\sqrt{2}}{2}$$. 6. Divide numerator terms by 2: $$1 \pm \sqrt{2}$$. 7. Therefore, the simplified expression has two values: $$1 + \sqrt{2}$$ and $$1 - \sqrt{2}$$. This completes the simplification.