Transcript Quadratic Equation

The Quadratic Formula
by Zach Barr
 b  (b  4ac)
2a
2
Simulation Online
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.
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1.
2.
For more information about the history of the Quadratic Formula, visit these two sites
The Original Problem
Babylonians
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.
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.
What to do?
ax 2  bx  c  0
 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.
Example: x  2 x  8  0
2
 For this equation: a=1, b=2, c=-8
 Use Quadratic formula:
 b  (b 2  4ac)
2a
 Plug in a, b, c to get:
 2  [( 2) 2  4(1)( 8)]
2(1)
Continuing Example
 2  [( 2)  4(1)( 8)]
2(1)
2
 Solve:
 2  4  32

2
26

2
x2
OR
OR
OR
 2  4  32

2
26

2
x  4
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
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.
The End!
By Zach Barr
9/26/2005