2. The Quadratic Formula by Zach Barr

3. Simulation Online

4. History of Quadratic Formula  A quadratic equation is a second order, univariate polynomial with constant coefficients and can usually be written in the form: ax^2 + bx + c = 0, where a cannot equal 0. In about 400 B.C. the Babylonians developed an algorithmic approach to solving problems that give rise to a quadratic equation. This method is based on the method of completing the square. Quadratic equations, or polynomials of second-degree, have two roots that are given by the quadratic formula: x = (-b +/- (b^2 - 4ac))/2a.  The earliest solutions to quadratic equations involving an unknown are found in Babylonian mathematical texts that date back to about 2000 B.C.. At this time the Babylonians did not recognize negative or complex roots because all quadratic equations were employed in problems that had positive answers such as length.  For more information about the history of the Quadratic Formula, visit these two sites 1.The Original Problem The Original ProblemThe Original Problem 2.Babylonians Babylonians

5. Deriving the Quadratic Formula  To derive the equation x^2 + bx + c = 0 into the quadratic formula, you must complete the square as shown on the webpage. webpage

6. What is the formula used for?  The Quadratic Formula is used to find the zeroes of an equation.  X represents the variable that we are trying to find. Because the equation is a second-order polynomial equation, with the term x^2, there will be two solutions.

7. What to do?  Given equation:  a, b, c are coefficients for the equation  Substitute in each value of a, b, c into the quadratic formula and solve for x.  Note: Remember the + OR – in front of the square root sign. This is how you get your two answers as you will see later.

8. Example:  For this equation: a=1, b=2, c=-8  Use Quadratic formula:  Plug in a, b, c to get:

9. Continuing Example  Solve: OR

10.  Use a graphing calculator to graph the equation x^2 +2x-8=0.  From the look of the graph, we find that the same two zeroes x=2,-4 as we did using the quadratic formula.

11. The End! By Zach Barr 9/26/2005