1. **State the problem:** Solve the quadratic equation $x^2 + 2x - 6 = 0$ for $x$. 2. **Formula used:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 1$, $b = 2$, and $c = -6$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4 \times 1 \times (-6) = 4 + 24 = 28$$ 5. **Apply the quadratic formula:** $$x = \frac{-2 \pm \sqrt{28}}{2 \times 1} = \frac{-2 \pm \sqrt{28}}{2}$$ 6. **Simplify the square root:** $$\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$$ 7. **Substitute back:** $$x = \frac{-2 \pm 2\sqrt{7}}{2}$$ 8. **Cancel common factor 2 in numerator and denominator:** $$x = \frac{\cancel{2}(-1 \pm \sqrt{7})}{\cancel{2}} = -1 \pm \sqrt{7}$$ 9. **Final solutions:** $$x_1 = -1 + \sqrt{7}$$ $$x_2 = -1 - \sqrt{7}$$ These are the two roots of the quadratic equation.