1. **State the problem:** Solve the quadratic equation $ax^2 + bx + c = 0$ for $x$. 2. **Formula used:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula finds the roots of any quadratic equation. 3. **Important rules:** - The term under the square root, $b^2 - 4ac$, is called the discriminant. - If the discriminant is positive, there are two real roots. - If it is zero, there is one real root (a repeated root). - If it is negative, the roots are complex. 4. **Example:** Solve $2x^2 - 4x - 6 = 0$. 5. **Calculate the discriminant:** $$\Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$ 6. **Apply the quadratic formula:** $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ 7. **Find the roots:** $$x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3$$ $$x_2 = \frac{4 - 8}{4} = \frac{\cancel{4} - 8}{\cancel{4}} = \frac{-4}{1} = -1$$ 8. **Answer:** The solutions are $x = 3$ and $x = -1$.