1. Problem: Find the value of $q$ in the quadratic equation $$x^2 - 8x + q = 0$$ given that the difference between its roots is 2. 2. Formula: For a quadratic equation $$ax^2 + bx + c = 0$$, the roots are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. Important rule: The difference between the roots $$x_1$$ and $$x_2$$ is $$|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}$$. 4. Apply the formula for the difference of roots: $$|x_1 - x_2| = \frac{\sqrt{(-8)^2 - 4 \cdot 1 \cdot q}}{1} = 2$$ 5. Simplify inside the square root: $$\sqrt{64 - 4q} = 2$$ 6. Square both sides to remove the square root: $$64 - 4q = 4$$ 7. Solve for $q$: $$64 - 4q = 4$$ $$\Rightarrow -4q = 4 - 64$$ $$\Rightarrow -4q = -60$$ $$\Rightarrow q = \frac{-60}{-4} = 15$$ 8. Final answer: $q = 15$. Therefore, the correct choice is B) 15.