1. **State the problem:** Solve the quadratic equation $x^2 + 12x - 8 = 0$ using the quadratic formula. 2. **Quadratic formula:** For any quadratic equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Important: The discriminant $\Delta = b^2 - 4ac$ determines the nature of roots. 3. **Identify coefficients:** Here, $a=1$, $b=12$, $c=-8$. 4. **Calculate discriminant:** $$\Delta = 12^2 - 4 \times 1 \times (-8) = 144 + 32 = 176$$ 5. **Apply quadratic formula:** $$x = \frac{-12 \pm \sqrt{176}}{2 \times 1} = \frac{-12 \pm \sqrt{176}}{2}$$ 6. **Simplify $\sqrt{176}$:** $$\sqrt{176} = \sqrt{16 \times 11} = 4\sqrt{11}$$ 7. **Substitute back:** $$x = \frac{-12 \pm 4\sqrt{11}}{2}$$ 8. **Simplify fraction by dividing numerator and denominator by 2:** $$x = \frac{\cancel{-12} \pm \cancel{4}\sqrt{11}}{\cancel{2}} = -6 \pm 2\sqrt{11}$$ 9. **Final solutions:** $$x_1 = -6 + 2\sqrt{11}, \quad x_2 = -6 - 2\sqrt{11}$$