1. **State the problem:** Solve the quadratic equation $$x^2 - 10x + 22 = 0$$. 2. **Identify coefficients:** Here, $$a = 1$$, $$b = -10$$, and $$c = 22$$. 3. **Recall the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula gives the solutions for any quadratic equation $$ax^2 + bx + c = 0$$. 4. **Substitute the values:** $$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(22)}}{2(1)}$$ 5. **Simplify inside the square root:** $$x = \frac{10 \pm \sqrt{100 - 88}}{2}$$ 6. **Calculate the discriminant:** $$100 - 88 = 12$$ 7. **Write the expression with the simplified discriminant:** $$x = \frac{10 \pm \sqrt{12}}{2}$$ 8. **Simplify the square root:** $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$ 9. **Substitute back:** $$x = \frac{10 \pm 2\sqrt{3}}{2}$$ 10. **Simplify the fraction by canceling 2:** $$x = \frac{\cancel{2} \times 5 \pm \cancel{2} \times \sqrt{3}}{\cancel{2}} = 5 \pm \sqrt{3}$$ **Final answer:** $$x = 5 + \sqrt{3} \quad \text{or} \quad x = 5 - \sqrt{3}$$