1. **State the problem:** We are given the quadratic equation $$3x^2 - 10x - 8 = 0$$ with roots $$\alpha$$ and $$\beta$$, where $$\alpha > \beta$$. We need to find the value of $$\alpha - \beta$$. 2. **Recall the quadratic formula:** The roots of $$ax^2 + bx + c = 0$$ are given by $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $$a=3$$, $$b=-10$$, and $$c=-8$$. 3. **Calculate the discriminant:** $$ \Delta = b^2 - 4ac = (-10)^2 - 4 \times 3 \times (-8) = 100 + 96 = 196 $$ 4. **Find the roots:** $$ \alpha = \frac{-b + \sqrt{\Delta}}{2a} = \frac{10 + 14}{6} = \frac{24}{6} = 4 $$ $$ \beta = \frac{-b - \sqrt{\Delta}}{2a} = \frac{10 - 14}{6} = \frac{-4}{6} = -\frac{2}{3} $$ 5. **Calculate $$\alpha - \beta$$:** $$ \alpha - \beta = 4 - \left(-\frac{2}{3}\right) = 4 + \frac{2}{3} = \frac{12}{3} + \frac{2}{3} = \frac{14}{3} $$ **Final answer:** $$\alpha - \beta = \frac{14}{3}$$