How to derive the general formula for solving quadratic equations

To derive the general formula for solving quadratic equations, follow these steps: 1. Start with the standard form of a quadratic equation: ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. 2. Divide both sides by 'a' to make the coefficient of x² equal to 1: x² + (b/a)x + (c/a) = 0. 3. Move the constant term to the right-hand side: x² + (b/a)x = -c/a. 4. Add (b/2a)² to both sides to complete the square: x² + (b/a)x + (b/2a)² = (b/2a)² - c/a. 5. Factor the left-hand side as a perfect square: (x + b/2a)² = (b/2a)² - c/a. 6. Take the square root of both sides: x + b/2a = ±√((b/2a)² - c/a). 7. Simplify the right-hand side: x + b/2a = ±√(b² - 4ac)/2a. 8. Solve for x by subtracting b/2a from both sides: x = (-b ± √(b² - 4ac))/2a. This is the general formula for solving quadratic equations, known as the quadratic formula: x = (-b ± √(b² - 4ac))/2a Note: This formula works for all quadratic equations, and the ± symbol indicates that there may be two solutions for x.