1. **State the problem:** Solve the quadratic equation $$x^2 - 10x + 34 = 0$$ for $x$. 2. **Formula used:** The quadratic formula for solving $ax^2 + bx + c = 0$ is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-10$, and $c=34$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-10)^2 - 4 \times 1 \times 34 = 100 - 136 = -36$$ Since the discriminant is negative, the solutions are complex (not real). 4. **Find the roots using the quadratic formula:** $$x = \frac{-(-10) \pm \sqrt{-36}}{2 \times 1} = \frac{10 \pm \sqrt{-36}}{2} = \frac{10 \pm 6i}{2}$$ 5. **Simplify the solutions:** $$x = 5 \pm 3i$$ **Final answer:** $$x = 5 + 3i, 5 - 3i$$