The General Solution to a Cubic Equation

According to “Maxima”, or more specifically, according to “wxMaxima”, the three Roots to a Cubic Equation are generally as shown below, assuming that there exists one solution entirely in Real numbers:

http://dirkmittler.homeip.net/cubic.pdf

(Edit 2/7/2016 : ) There are two observations which need to be made about the solution shown above, which are related to the fact that a cubic equation can sometimes have three Real roots, or two, but that it always has at least one.

1) The expression which we’re told to find the cube root of could be equal to zero. And while finding the cube root of zero represents no obstacle, a division by zero does, and a division by zero ensues.

2) The expression we’re asked to find the square root of can become negative. In that case the solution shown above finds no Real numbers. Further, this output from ‘Maxima’ does not elucidate, how to process the fact that radicals are usually both negative and positive. An entire expression gets repeated, in which the radical could be negative. And there is no easy way to know, whether this radical is allowed to be negative in only one occurrence, or in both occurrences…

When using ‘Maxima’, a frequent goal is to eliminate extraneous complex numbers, by applying the sequence [‘rectform’, ‘trigsimp’] to an already-formed solution which is capable of producing Real numbers. But in this example, the sequence does not produce meaningful results. And one main reason is the fact that this sequence has no magic, by which to output information which was not input. So this trick does not produce an inverse-trigonometric function whose angle is naturally divided by three, so that a multiple of (2π/3) Radians could simply be added to it, before a trig function is taken again. That ‘Maxima’ can recognize.

(End of Edit 2/7/2016)

If we need to find three existing real roots, then we must apply the system of Reduction To A Depressed Cubic as shown here:

Step 1

Followed by Trigonometric Method For Three Real Roots as shown here:

Step 2

 

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My Site was down early this morning.

I use my computers at home to host my Web-site, including this blog. But the IP address I have at home, is just a regular, dynamically-assigned IP address, which is owned by my ISP. This IP address can be reassigned to be at any moment.

Therefore, I have an update service which makes my changed IP address available to people who may want to access my URLs.

This morning, this update service failed to do its job correctly. And therefore, my Site was unavailable from 4h40 until 6h50.

This problem has now been corrected in an easy way.

Dirk

 

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Switched In a Replacement Keyboard

This is a situation which brings back memories of my Late Father, who repatriated to Germany in the late 1990s and early 2000s. He had a habit of giving me gifts that included computers and computer accessories.

During a visit which I made to Germany in 2010, he gave me a laptop that he didn’t need anymore, and which is in fact the Acer Aspire 5020 I mentioned before.

Link to Previous Posting

But his gifts that year included a good laptop bag, and also a keyboard. What I found peculiar about Dad, was that it had seemed very important to him, to find a keyboard in Germany which emulated the standard 105-key, US keyboard layout. Evidently he had been away from Canada and the USA for so long, that he could not remember what the US keyboard layout was. And so specifically for me, he had purchased a new K.B., which had the U.K. layout. This is entirely logical, because of the geographic proximity Germany has to Great Britain.

But Dad seemed struck so sad, when he learned that the K.B. he had bought for me wasn’t really a US-layout board. I tried to explain to him, that a Linux computer can easily be switched from one keyboard layout to another, and that the only challenge we faced, was to identify which layout this keyboard had. Because, we had actually failed to find this out, for which reason I did not make immediate use of it.

And, we did not have the time to solve the puzzle either, because we had his laptop to set up, as well as numerous other things to do in Germany, while my Father was still alive.

But, Not knowing what the keyboard layout is, can do far more damage, than simply having a layout from a nearby country. But Dad felt it was the other way around.

The computer I name ‘Pheonix’ possessed a Hewlett-Packard keyboard which had always served me well. But now the time came to retire that old K.B., which was my occasion this evening to switch in the one which my Late Father had given to me in 2010.

This K.B. is of the brand-name “Cherry”, and just by Googling that, I found out that Cherry was a later generation, high-quality keyboard brand. I tend to appreciate keyboards that have good tactile properties, and high-quality switches. Even though Cherries are not genuine “Clickety Keyboards”, they come close. I expect that this one will last me a long time. It’s quite robust.

Dirk

 

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