1. **Stating the problem:** The user requests the most detailed step-by-step explanation for a math problem, but no specific problem is given. 2. Since no explicit problem is provided, I will demonstrate a detailed step-by-step solution for a common algebra problem: solving the quadratic equation $ax^2 + bx + c = 0$. 3. **Formula used:** The quadratic formula to find roots of $ax^2 + bx + c = 0$ is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Important rules:** - The discriminant $\Delta = b^2 - 4ac$ determines the nature of roots. - If $\Delta > 0$, two distinct real roots. - If $\Delta = 0$, one real root (repeated). - If $\Delta < 0$, two complex roots. 5. **Example:** Solve $2x^2 - 4x - 6 = 0$. 6. Identify coefficients: $a=2$, $b=-4$, $c=-6$. 7. Calculate discriminant: $$\Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$ 8. Since $\Delta = 64 > 0$, there are two distinct real roots. 9. Calculate roots: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ 10. Compute each root: - $x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3$ - $x_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1$ 11. **Final answer:** The solutions to $2x^2 - 4x - 6 = 0$ are $x=3$ and $x=-1$. This detailed explanation covers the problem statement, formula, rules, intermediate steps, and final solution.