1. **State the problem:** Solve the quadratic equation $x^2 - 4x + 8 = 0$. 2. **Recall the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-4$, and $c=8$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 1 \times 8 = 16 - 32 = -16$$ Since $\Delta < 0$, the equation has no real roots but two complex roots. 4. **Find the roots using the quadratic formula:** $$x = \frac{-(-4) \pm \sqrt{-16}}{2 \times 1} = \frac{4 \pm \sqrt{-16}}{2} = \frac{4 \pm 4i}{2}$$ 5. **Simplify the roots:** $$x = 2 \pm 2i$$ **Final answer:** The solutions to the equation are $x = 2 + 2i$ and $x = 2 - 2i$.