Understanding the Quadratic Formula and Graphical Solutions

TLDR; The quadratic formula is used to solve equations and verify solutions graphically, with a connection to the x-intercepts of quadratic functions.

📈 Understanding the Quadratic Formula

The quadratic formula provides solutions to any quadratic equation of the form ax^2 + bx + c = 0, where 'a' does not equal 0.

The formula, x = (-b ± √(b^2 - 4ac)) / (2a), is derived by completing the square on the equation ax^2 + bx + c = 0.

It is not a magical formula but a result of a specific method of solving quadratic equations.

📊 Connection to Graphical Solutions

The zeros of a quadratic function occur where y or f(x) = 0, and if they are real, they correspond to the x-intercepts of the parabola.

The x-intercepts of a quadratic function are connected to the solutions of quadratic equations, as they represent the real values of the solutions.

✏️ Example: Solving a Quadratic Equation

An example of solving a quadratic equation, -x^2 + 3x + 10 = 0, is provided using the quadratic formula and then verified graphically.

The values of 'a', 'b', and 'c' are identified, substituted into the formula, and then simplified to obtain the solutions.

The graphical verification confirms the correctness of the solutions by showing the x-intercepts of the parabola.

🔍 Another Example: Verification Graphically

Another example, 2x^2 - 4x - 3 = 0, is solved using the quadratic formula and its solutions are verified graphically.

The process involves substitution, simplification, and then the graphical verification of the x-intercepts on a graphing calculator.