Help required in understanding few parameters of cubic equation formula for solving polynomial of degree three.

[Gevni]

Hi, Can someone please help me in understanding few parameters of cubic equation formula for solving polynomial of degree three. I attached the formula in the screenshot.
My questions are:
(1) what is ". " dot in the end of the formula and what does it mean?
(2) I want to use it only for real number not complex, so do the formula remain the same?
(3) It states the solution is for this kind of equation [MATH] ax^3+ax^2+cx+d=0 [/MATH] what if I have equation like that [MATH]ax^3+ax^2-cx+d=0 [/MATH]? Do i have to put negative value with c parameter in the below formula or whatever the case formula remain same?
(4) It is just one formula then how I can get three roots by using this?
(5) I want to use this formula for getting the minimum possible value of x and if it results in 3 roots then which root show minimum value of x.

I am very thankful if someone help me understanding this formula or any resource that can help me understanding this.







View attachment 9772

 

[Gevni]

UPDATE: Please consider this one as previously, I attached the wrong image
Hi, Can someone please help me in understanding few parameters of cubic equation formula for solving polynomial of degree three. I attached the formula in the screenshot.
My questions are:
(1) what is ". " dot in the end of the formula and what does it mean?
(2) I want to use it only for real number not complex, so do the formula remain the same?
(3) It states the solution is for this kind of equation [MATH] ax^3+ax^2+cx+d=0 [/MATH] what if I have equation like that [MATH]ax^3+ax^2-cx+d=0 [/MATH]? Do i have to put negative value with c parameter in the below formula or whatever the case formula remain same?
(4) It is just one formula then how I can get three roots by using this?
(5) I want to use this formula for getting the minimum possible value of x and if it results in 3 roots then which root show minimum value of x.

I am very thankful if someone help me understanding this formula or any resource that can help me understanding this.

 

[Dr.Peterson]

(1) The dot at the end of each formula is a period marking the end of a sentence. They sometimes bother me, too.

(2) A cubic equation can have one, two, or three real solutions. This formula gives, not "the solution" as it inadequately says, but "a real solution". If there are other real solutions, you have to find them separately.

(3) If c is replaced by -c in the equation, just do the same in the formula. That is, if a term is -5x, then take c as -5, not 5.

(4) You can get the other two roots by dividing the cubic by the factor corresponding to the root you found, which will yield a quadratic equation that you can solve by the formula you already know. Alternatively, you could find all three cube roots, rather than just the principal roots, using the complex cube roots of unity. (With this method, you can't avoid complex numbers even in the best case!)

(5) Compare the three roots you get (if they are all real).
I found that it comes from here: https://math.vanderbilt.edu/schectex/courses/cubic/

You will note that this paper points out later that it will sometimes involve calculations with imaginary numbers, even though the final result is real. This is unavoidable.

You will also note that the author is intentionally directing you away from using this formula, for that reason and others; this is probably why he does not clearly state that this is not "the solution", but "the easy solution" -- the others take more work!

I have to admit that I have never done much with this formula, even in the usual form in which it is broken into pieces, so I might have a couple details wrong.

 

[Gevni]

(1) The dot at the end of each formula is a period marking the end of a sentence. They sometimes bother me, too.

(2) A cubic equation can have one, two, or three real solutions. This formula gives, not "the solution" as it inadequately says, but "a real solution". If there are other real solutions, you have to find them separately.

(3) If c is replaced by -c in the equation, just do the same in the formula. That is, if a term is -5x, then take c as -5, not 5.

(4) You can get the other two roots by dividing the cubic by the factor corresponding to the root you found, which will yield a quadratic equation that you can solve by the formula you already know. Alternatively, you could find all three cube roots, rather than just the principal roots, using the complex cube roots of unity. (With this method, you can't avoid complex numbers even in the best case!)

(5) Compare the three roots you get (if they are all real).
I found that it comes from here: https://math.vanderbilt.edu/schectex/courses/cubic/

You will note that this paper points out later that it will sometimes involve calculations with imaginary numbers, even though the final result is real. This is unavoidable.

You will also note that the author is intentionally directing you away from using this formula, for that reason and others; this is probably why he does not clearly state that this is not "the solution", but "the easy solution" -- the others take more work!

I have to admit that I have never done much with this formula, even in the usual form in which it is broken into pieces, so I might have a couple details wrong.

Thank you so much for your detailed reply. You clear almost all of my questions. I am not going to use this formula as I am not getting the right value of x. Now, I am trying to solve the equation by using Cardano’s formula. I am only interested in real value not the complex part.

 

[Dr.Peterson]

You can't avoid "the complex part"! In fact, my understanding is that this is why complex numbers were recognized: even when the solution is a real number, you sometimes have to, as I put it, "go through the complex world" in order to get to it. It's not that there is a complex part of the answer, but that getting to the answer, which is a real number, requires using complex numbers.

Not knowing the specifics of your goal, I can't be sure what method to recommend instead. Normally, if you are writing a program to solve a general cubic, you would use a numerical approximation method. If you want all three roots, if they exist, I don't know what is considered the most efficient way. Some others here may have more experience with that.

 

[Gevni]

My problem is solved. I wrote a C program for cubic equation using Cardano’s formula, that will only pick the roots when determinant >=0 and if squreroot is of any negative number it stop and only display results when the output is real number within the boundary and I got my formula result Many many thanks for your help, your reply really give me the right direction to work.