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.
