1. **State the problem:** Given the polynomial $$-x^2 + 8x + 9$$ with zeroes $$\alpha$$ and $$\beta$$ where $$\alpha > \beta$$, find the value of $$\alpha - \beta$$. 2. **Recall the formulas:** For a quadratic polynomial $$ax^2 + bx + c$$ with roots $$\alpha$$ and $$\beta$$, - Sum of roots: $$\alpha + \beta = -\frac{b}{a}$$ - Product of roots: $$\alpha \beta = \frac{c}{a}$$ 3. **Identify coefficients:** Here, $$a = -1$$, $$b = 8$$, and $$c = 9$$. 4. **Calculate sum and product:** - $$\alpha + \beta = -\frac{8}{-1} = 8$$ - $$\alpha \beta = \frac{9}{-1} = -9$$ 5. **Use the formula for difference of roots:** $$\alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta}$$ 6. **Substitute values:** $$\alpha - \beta = \sqrt{8^2 - 4 \times (-9)} = \sqrt{64 + 36} = \sqrt{100}$$ 7. **Simplify:** $$\alpha - \beta = 10$$ **Final answer:** $$\boxed{10}$$