## 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:

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.

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