1. **State the problem:** Solve the quadratic equation $$3x^2 + 5x - 2 = 0$$ to find the roots (values of $x$). 2. **Formula used:** For a quadratic equation $$ax^2 + bx + c = 0$$, the roots are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $a = 3$, $b = 5$, and $c = -2$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4 \times 3 \times (-2) = 25 + 24 = 49$$ 5. **Evaluate the roots:** $$x = \frac{-5 \pm \sqrt{49}}{2 \times 3} = \frac{-5 \pm 7}{6}$$ 6. **Find each root:** - For the plus sign: $$x = \frac{-5 + 7}{6} = \frac{2}{6} = \frac{1}{3}$$ - For the minus sign: $$x = \frac{-5 - 7}{6} = \frac{-12}{6} = -2$$ 7. **Final answer:** The roots are $$\left\{-2, \frac{1}{3}\right\}$$ which corresponds to option A. This means the parabola crosses the x-axis at $x = -2$ and $x = \frac{1}{3}$.