Paco Arjonilla: HP 35S Program Suite

 Paco Arjonilla:  HP 35S Program Suite

On today's blog post, I am going to highlight a recent suite of programs, written by Paco Arjonilla, is designed to compliment the rich function set of the HP 35S calculator.

Each category will have a set of programs of functions.  Stack diagrams are provided in the documentation.  Each category belongs to a single label.

Label P:  vector functions (includes norm, cross product, 3 x 3 determinant) - 149 steps

Label G:  complex numbers (conjugate, polar/rectangular conversions, construct and break down complex numbers) - 48 steps

Label Q:  quaternions (construction, angle, rotation) - 80 steps

Label K:  quadratic equation solver - 28 steps

Label H:  save stack data to variables - 55 steps

Label I:  recall saved stack - 55 steps

Label J:  stack operations (push and pop the x stack, delete the x stack, clear reserved memory registers, copy memory registers) - 159 steps

Paco's website also has a cheat sheet which can be downloaded as a PDF file.

For the programs and for license information, click here:  

http://www.pacoarjonilla.es/HP-35s.html

Special thanks to Paco Arjonilla.

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. 

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.