1. Problem: Find the roots of the quadratic equation $$3x^2 - 2x - 5 = 0$$ 2. Formula: The roots of a quadratic equation $$ax^2 + bx + c = 0$$ are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. Identify coefficients: Here, $$a=3$$, $$b=-2$$, and $$c=-5$$. 4. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 3 \times (-5) = 4 + 60 = 64$$ 5. Since $$\Delta > 0$$, there are two distinct real roots. 6. Calculate the roots: $$x = \frac{-(-2) \pm \sqrt{64}}{2 \times 3} = \frac{2 \pm 8}{6}$$ 7. Find each root: - $$x_1 = \frac{2 + 8}{6} = \frac{10}{6} = \frac{\cancel{10}}{\cancel{6}} = \frac{5}{3}$$ - $$x_2 = \frac{2 - 8}{6} = \frac{-6}{6} = \frac{\cancel{-6}}{\cancel{6}} = -1$$ Final answer: The roots are $$x = \frac{5}{3}$$ and $$x = -1$$.