1. **State the problem:** Solve a quadratic equation of the form $$ax^2 + bx + c = 0$$ where $a \neq 0$. 2. **Formula used:** The quadratic formula to find the roots is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Important rules:** - The term under the square root, $\Delta = b^2 - 4ac$, is called the discriminant. - If $\Delta > 0$, there are two distinct real roots. - If $\Delta = 0$, there is exactly one real root (a repeated root). - If $\Delta < 0$, the roots are complex (not real). 4. **Algorithm steps:** - Step 1: Identify coefficients $a$, $b$, and $c$ from the quadratic equation. - Step 2: Calculate the discriminant $\Delta = b^2 - 4ac$. - Step 3: Evaluate the square root $\sqrt{\Delta}$. - Step 4: Compute the two roots using the formula: $$x_1 = \frac{-b + \sqrt{\Delta}}{2a}$$ $$x_2 = \frac{-b - \sqrt{\Delta}}{2a}$$ - Step 5: Interpret the roots based on the discriminant value. 5. **Explanation:** This method works because the quadratic formula is derived by completing the square on the general quadratic equation. It guarantees finding all roots (real or complex) of any quadratic equation. This algorithm can be implemented in any programming language or solved by hand using these steps.