1. **State the problem:** Solve the quadratic equation $$x^2 + x - 12 = 0$$ for $x$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients. 3. **Identify coefficients:** Here, $a = 1$, $b = 1$, and $c = -12$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 1^2 - 4 \times 1 \times (-12) = 1 + 48 = 49$$ 5. **Find the square root of the discriminant:** $$\sqrt{\Delta} = \sqrt{49} = 7$$ 6. **Apply the quadratic formula:** $$x = \frac{-1 \pm 7}{2 \times 1} = \frac{-1 \pm 7}{2}$$ 7. **Calculate the two roots:** - For the plus sign: $$x = \frac{-1 + 7}{2} = \frac{6}{2} = 3$$ - For the minus sign: $$x = \frac{-1 - 7}{2} = \frac{-8}{2} = -4$$ 8. **Final answer:** The solutions to the equation are $$x = 3$$ and $$x = -4$$.