By Kardi Teknomo, PhD.

AHP

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Quadratic Equation

In this section of quadratic function, you will learn about quadratic equation, discriminant, and solving the equation through quadratic formula and modified quadratic formula that is better numerically.

Quadratic equation Quadratic Equation: Discriminant is happened to be a special condition of quadratic function Quadratic Equation: Discriminant when the parabola curve intersects with the horizontal axis ( Quadratic Equation: Discriminant ). In this case, we can three possibilities:

  1. The quadratic function does not intersect horizontal axis
  2. The quadratic function intersect horizontal axis at single point
  3. The quadratic function intersect horizontal axis at two points

Quadratic Equation: Discriminant

Quadratic Equation: Discriminant

Quadratic Equation: Discriminant

No intersection, Quadratic Equation: Discriminant

Single intersection, Quadratic Equation: Discriminant

Two intersections, Quadratic Equation: Discriminant

The point of intersection between the function and horizontal axis is called the zeros of the function, or the root of the quadratic equation, because inputting the Quadratic Equation: Discriminant values into the quadratic function will make the function value to be zero.

Discriminant

Discriminant is an index to indicate whether the quadratic function has no intersection ( Quadratic Equation: Discriminant ), intersect as single point ( Quadratic Equation: Discriminant ) or intersect at multiple roots ( Quadratic Equation: Discriminant ) with the horizontal axis. Discriminant of quadratic equation Quadratic Equation: Discriminant is computed through its parameters:

Quadratic Equation: Discriminant

When the discriminant is non-negative, given a quadratic equation Quadratic Equation: Discriminant , our problem is to solve the equation to find the value of Quadratic Equation: Discriminant that satisfied the equation. Solving quadratic equation is actually a process of finding the roots (or the zeros) of the equation.

As this is a well-known problem, the roots of quadratic equation (for multiple roots) are given by quadratic formula:

Quadratic Equation: Discriminant and Quadratic Equation: Discriminant

The version, which contains all three parameters of quadratic equation, is sometimes called ABC formula

Quadratic Equation: Discriminant

The roots of quadratic equation are always multiple roots (two roots) except when the discriminant is zero.

For example :

Quadratic equation Quadratic Equation: Discriminant contains parameter Quadratic Equation: Discriminant , Quadratic Equation: Discriminant and Quadratic Equation: Discriminant . The discriminant is Quadratic Equation: Discriminant . Since the discriminant is negative, we conclude that the equation has no real root (no intersection between the function with horizontal axis). However, the roots exist. The roots are multiple roots in complex number, given by the quadratic formula above

Quadratic Equation: Discriminant

Quadratic Equation: Discriminant

In his interesting paper, Baker (1998) pointed out that for quadratic equation Quadratic Equation: Discriminant (notice that parameter Quadratic Equation: Discriminant ), we can get Quadratic Equation: Discriminant and Quadratic Equation: Discriminant , therefore the quadratic formula above can be written as two terms

Quadratic Equation: Discriminant

The first term ( Quadratic Equation: Discriminant ) is mean or center of mass of the two roots and the second term

( Quadratic Equation: Discriminant ) is half of the absolute difference of the two roots.

In case of a single (distinct) root, the discriminant is zero, thus lead to

Quadratic Equation: Discriminant

Example

Quadratic equation Quadratic Equation: Discriminant has parameters Quadratic Equation: Discriminant and Quadratic Equation: Discriminant . The discriminant is zero because Quadratic Equation: Discriminant . Thus, the root of the quadratic equation is only one, that is Quadratic Equation: Discriminant . To check the correctness, we input the root value into the equation to get Quadratic Equation: Discriminant which is correct.

Another version of quadratic formula is given as

Quadratic Equation: Discriminant

Press et al (1986) pointed out that when the value of parameter Quadratic Equation: Discriminant and Quadratic Equation: Discriminant are both very small, using quadratic formulas above, the root of discriminant will be inaccurate numerically because Quadratic Equation: Discriminant and Quadratic Equation: Discriminant might cancel each other. They proposed formula that is more robust in term of numerical accuracy

Quadratic Equation: Discriminant

Quadratic Equation: Discriminant and Quadratic Equation: Discriminant

Example

Suppose we have a quadratic equation Quadratic Equation: Discriminant . We can divide two sides of the equation with 2 to get the equivalent equation of Quadratic Equation: Discriminant . Note that now the quadratic parameter is one ( Quadratic Equation: Discriminant ). The other parameters are Quadratic Equation: Discriminant and Quadratic Equation: Discriminant . The discriminant is Quadratic Equation: Discriminant that is positive. Thus, the quadratic equation has multiple roots

Quadratic Equation: Discriminant

Quadratic Equation: Discriminant

Using Press et al modified Quadratic formula we get the same answers

Quadratic Equation: Discriminant

Quadratic Equation: Discriminant

Quadratic Equation: Discriminant

Observe that Quadratic Equation: Discriminant and

Quadratic Equation: Discriminant

So far, we have covered only a given quadratic function. In the next section , you will learn how to get the quadratic parameters from data points.

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These tutorial is copyrighted .

Preferable reference for this tutorial is

Teknomo, Kardi. (2008) Quadratic Function Tutorial .
http://people.revoledu.com/kardi/tutorial/quadratic/