Sunday, September 18, 2022

Quick Tip: Determining the Characteristics of a Quadratic Equation

Quick Tip:   Determining the Characteristics of a Quadratic Equation


Introduction


For our algebra and college pre-calculus students, here is a quick way to tell whether the quadratic equations 


A * x^2 + B * x + C = 0


has real roots or complex roots (in the form of a + bi or r*e^(Θi)).  The simple way is to calculate the discriminant.


The discriminant of the quadratic equation is B^2 - 4 * A * C.   


If B^2 > 4 * A * C, or B^2 - 4 * A * C > 0, the roots are real and distinct


If B^2 = 4 * A * C, or B^2 - 4 * A * C = 0, there is a repeated root


If B^2 < 4 * A * C, or B^2 - 4 * A * C < 0, the roots are complex


(A, B, C are real numbers)



Examples


3 * x^2 - 6 * x + 81 = 0

B^2 = 36

4 * A * C = 972

36 < 972

The roots are complex  (1 ± i√26)


4 * x^2 + 44 * x - 318 = 0

B^2 = 1936

4 * A * C = -5088

1936 > -5088

The roots are real and distinct ( (-11 ±√439)/2 )


-3 * x^2 - 6 * x - 5

B^2 = 6

4 * A * C = 60

6 < 60

The roots are complex ( (-3 ± i√6)/ 3)



A Study 


If we let A = 1 and B and C range of integers through -5 to 5, if we pick a quadratic equation from random we find that:


25.62% of the equations have complex roots

4.13% has a repeated root

70.25% has two distinct real roots


Here is the Google Sheet that has the study:  

https://docs.google.com/spreadsheets/d/1ZKAR1dtnHAss1CzxqygHCIB3Mq2u2fn3TLR3espUUXM/edit?usp=sharing



Hope this helps,


Eddie 


All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


Matrices in Python without Numpy: Part 1

Matrices in Python without Numpy:  Part 1 Introduction Python is a wonderful programming language and is a welcome addition to graphing calc...