## 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: