1. **Problem Statement:** Solve a quadratic equation of the form $$ax^2 + bx + c = 0$$ where $a \neq 0$. 2. **Formula Used:** The quadratic formula to find roots is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Step-by-step Explanation:** - Calculate the discriminant $$\Delta = b^2 - 4ac$$. - If $$\Delta > 0$$, there are two distinct real roots. - If $$\Delta = 0$$, there is exactly one real root (a repeated root). - If $$\Delta < 0$$, there are two complex roots. 4. **Intermediate Work:** - Compute $$\sqrt{\Delta}$$. - Substitute values into the formula: $$x_1 = \frac{-b + \sqrt{\Delta}}{2a}$$ $$x_2 = \frac{-b - \sqrt{\Delta}}{2a}$$ 5. **Learner-friendly Explanation:** - First, identify coefficients $a$, $b$, and $c$ from the quadratic equation. - Then, find the discriminant to understand the nature of the roots. - Use the quadratic formula to calculate the roots based on the discriminant. - This method works for all quadratic equations and gives exact solutions. This flowchart approach guides you through these steps logically to solve any quadratic equation.