1. **State the problem:** Solve the quadratic equation $$5x^2 - 3x + 2 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = 5$$, $$b = -3$$, and $$c = 2$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-3)^2 - 4 \times 5 \times 2 = 9 - 40 = -31$$. 5. **Interpret the discriminant:** Since $$\Delta < 0$$, there are no real roots; the solutions are complex numbers. 6. **Find the roots:** $$x = \frac{-(-3) \pm \sqrt{-31}}{2 \times 5} = \frac{3 \pm \sqrt{-31}}{10} = \frac{3 \pm i\sqrt{31}}{10}$$ 7. **Final answer:** $$x = \frac{3}{10} + \frac{i\sqrt{31}}{10} \quad \text{or} \quad x = \frac{3}{10} - \frac{i\sqrt{31}}{10}$$