1. Let's start by understanding the problem: you want a step-by-step explanation for a math problem, but you haven't specified which problem. 2. Since no specific problem was given, I'll demonstrate a general approach to solving a quadratic equation step-by-step. 3. Consider the quadratic equation $ax^2 + bx + c = 0$. The formula to find the roots is given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. Important rules: - The discriminant $\Delta = b^2 - 4ac$ determines the nature of the roots. - If $\Delta > 0$, there are two distinct real roots. - If $\Delta = 0$, there is one real root (a repeated root). - If $\Delta < 0$, the roots are complex. 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 the equation $2x^2 - 4x - 6 = 0$ are $x=3$ and $x=-1$.