1. **State the problem:** We are given the quadratic equation $$x^2 - 2x - 9 = 0$$ and told one solution can be written as $$1 + \sqrt{k}$$. We need to find the value of $$k$$. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a=1$$, $$b=-2$$, and $$c=-9$$. 4. **Calculate the discriminant:** $$b^2 - 4ac = (-2)^2 - 4(1)(-9) = 4 + 36 = 40$$ 5. **Find the roots using the quadratic formula:** $$x = \frac{-(-2) \pm \sqrt{40}}{2(1)} = \frac{2 \pm \sqrt{40}}{2} = 1 \pm \frac{\sqrt{40}}{2}$$ 6. **Rewrite the root in the form $$1 + \sqrt{k}$$:** We want to express $$1 + \frac{\sqrt{40}}{2}$$ as $$1 + \sqrt{k}$$. 7. **Equate and solve for $$k$$:** $$\sqrt{k} = \frac{\sqrt{40}}{2} = \sqrt{\frac{40}{4}} = \sqrt{10}$$ Thus, $$k = 10$$. **Final answer:** $$k = 10$$ (Option B).