1. The problem is to find the roots of the quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$. 2. The formula used is the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $b^2 - 4ac$ is the discriminant. 3. Important rules: - If the discriminant is positive, there are two distinct real roots. - If the discriminant is zero, there is one real root (a repeated root). - If the discriminant is negative, there are two complex roots. 4. Example: Solve $2x^2 - 4x - 6 = 0$. 5. Calculate the discriminant: $$\Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$ 6. Since $\Delta > 0$, there are two real roots. 7. Apply the quadratic formula: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ 8. Calculate the roots: - $$x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3$$ - $$x_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1$$ 9. Final answer: The roots are $x = 3$ and $x = -1$. This process can be applied to any quadratic equation to find its roots.