1. **State the problem:** Solve the quadratic equation $$3x^2 - 5x + 2 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the equation is in the form $$ax^2 + bx + c = 0$$. 3. **Identify coefficients:** Here, $$a = 3$$, $$b = -5$$, and $$c = 2$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 3 \times 2 = 25 - 24 = 1$$. 5. **Apply the quadratic formula:** $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 3} = \frac{5 \pm 1}{6}$$. 6. **Find the two solutions:** - For the plus sign: $$x = \frac{5 + 1}{6} = \frac{6}{6} = 1$$. - For the minus sign: $$x = \frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3}$$. 7. **Simplify the fraction:** $$\frac{4}{6} = \frac{\cancel{2} \times 2}{\cancel{2} \times 3} = \frac{2}{3}$$. **Final answer:** $$x = 1 \text{ or } x = \frac{2}{3}$$.