1. **State the problem:** Solve the quadratic equation $$2x^2 + 3x - 20 = 0$$. 2. **Formula and rules:** To solve a quadratic equation of the form $$ax^2 + bx + c = 0$$, we can use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=3$, and $c=-20$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4 \times 2 \times (-20) = 9 + 160 = 169$$ 4. **Apply the quadratic formula:** $$x = \frac{-3 \pm \sqrt{169}}{2 \times 2} = \frac{-3 \pm 13}{4}$$ 5. **Find the two solutions:** - For the plus sign: $$x = \frac{-3 + 13}{4} = \frac{10}{4} = \frac{\cancel{10}}{\cancel{4}} = \frac{5}{2} = 2.5$$ - For the minus sign: $$x = \frac{-3 - 13}{4} = \frac{-16}{4} = \frac{\cancel{-16}}{\cancel{4}} = -4$$ 6. **Final answer:** The solutions to the equation are $$x = 2.5$$ and $$x = -4$$.