1. **State the problem:** Solve the quadratic equation $$3x^2 + 5x - 12 = 0$$. 2. **Formula used:** For a quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = 3$$, $$b = 5$$, and $$c = -12$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4 \times 3 \times (-12) = 25 + 144 = 169$$ 5. **Find the square root of the discriminant:** $$\sqrt{169} = 13$$ 6. **Apply the quadratic formula:** $$x = \frac{-5 \pm 13}{2 \times 3} = \frac{-5 \pm 13}{6}$$ 7. **Calculate the two solutions:** - For the plus sign: $$x = \frac{-5 + 13}{6} = \frac{8}{6} = \frac{4}{3}$$ - For the minus sign: $$x = \frac{-5 - 13}{6} = \frac{-18}{6} = -3$$ **Final answer:** The solutions to the equation are $$x = \frac{4}{3}$$ and $$x = -3$$.