1. We are asked to solve the quadratic equation $$x^2 + 2x - 6 = 0$$ for $x$. 2. The quadratic formula to find roots of $$ax^2 + bx + c = 0$$ is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=2$, and $c=-6$. 3. Calculate the discriminant: $$\Delta = b^2 - 4ac = 2^2 - 4 \times 1 \times (-6) = 4 + 24 = 28$$ 4. Substitute into the quadratic formula: $$x = \frac{-2 \pm \sqrt{28}}{2 \times 1} = \frac{-2 \pm \sqrt{28}}{2}$$ 5. Simplify the square root: $$\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$$ 6. Substitute back: $$x = \frac{-2 \pm 2\sqrt{7}}{2}$$ 7. Cancel the common factor 2 in numerator and denominator: $$x = \frac{\cancel{2}(-1 \pm \sqrt{7})}{\cancel{2}} = -1 \pm \sqrt{7}$$ 8. Final solutions: $$x = -1 + \sqrt{7} \quad \text{and} \quad x = -1 - \sqrt{7}$$ These are the two roots of the quadratic equation.