Saturday, 16 August, 2008

How To Identify Type Of Roots In Quadratic Equation

If we are asked to solve mathematical equations that equal to zero, the objective is to find their roots.

Roots are those numerial answers to the variable in the math equations.

Solve x2+2x+1 = 0.

x2+2x+1 = (x - 1)2 = 0
This gives us x = 1 as the answer.
This x = 1 is the ROOT to the equation.

However, we need to know there are 3 types of roots to a equation.

They are:

  1. Real and different

  2. Real and equal

  3. Imaginary

How do we know or identify the roots given an equation?

One way to identify is through graph sketching and identifying the type of roots through the interception of the curve with the y=0 line (or x-axis).

Example A:
From the graph plotted (diagram 1), what is the type of root?

Diagram 1

From diagram 1, through observing the graph, we noticed the 2 red "x" markings that the curve makes with the y= 0 line (or the horizontal x-axis). These are the 2 roots of the equation (not known here) used to plot the graph.

The 2 roots in diagram 1, can be said to be Real and also Different, since they made true numerical values that can be readable by anyone understanding graph.

Example B:
What is the type of roots in diagram 2?

Diagram 2

In diagram 2, we see that the curve made only 1 point of contact with the x-axis or y = o line. Therefore, there is only one Real root for this equation used to plot the graph.

Example C:
How about the type of roots for this graph (diagram 3)?

Diagram 3

Diagram 3 is a bit special in the sense that we cannot see any root cutting the x-axis.

This, therefore, calls for imagination !

To solve equation that produce this type of "floating" curve, special roots called Imaginary roots are conceptualised. They do not exist and are thus un-real.

In short, when we see graph having no x-axis crossing by the curve, the roots are of type Imaginary .

The above is using graphical method to identify types of root in an equation.

Another method is through the famous "Quadratic Formula".

However, we do not require the full formula to identify the root type.

What we need is just the "(b2-4ac)" portion of the formula. This is sufficient to extract information to pinpoint the type of roots in any given math quadratic equation.

Note: The general quadratic equation is ax2 + bx + c2.

The demerit of this "Quadratic Formula" approach to identify types of root is limited to Quadratic equation. Equation in the order of 3 and above CANNOT use this approach and graphical method stand an advantage in this case.

How to use this Quadratic Formula approach to identify the root type?

If the numerical value of the "(b2-4ac)" is :

  • > 0 ==> The roots are Real and Different type

  • = 0 ==> The root is Real and Equal type

  • < 0 ="="> The roots are Imaginary type.

The type is logically derived as a result of square-rooting the "(b2-4ac)" .

The principle of graph reading and interpreting its data yields many useful information and applications. Graph can, therefore, complement other method of mathematical analysis to solve problems, and can at times be even simpler.


1 comment:

Anonymous said...

The general quadratic equation is
a x^2 + b x + c

c is not squared because it is the y-intercept.

By graphing, the y-intercept is not c^2.