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
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