Cubic Formula Puns

Learnt about the quadratic formula and how it’s derived and wanted to know if there is some way to get a similar formula for cubic equations by completing a cube instead of a square. I want to try to figure it out on my own so if anyone can provide hints on where to start like what identities to use etc that would help. On the other hand if this method doesn’t work or is ridiculously hard to figure out please do warn me. Thank you. Also if it is possible for me to derive without too much suffering pls don't link to the actual formula ty.

For example, how do I find what n³ is given (n-1)³?

I know a method for square numbers. Finding 23² using the previous squared number is 22² + 22 + 23. In general, n² = (n-1)² + (n-1) + n

I just want to find a similar method for cubic numbers. Thanks for anyone who helps!

Edit: I would like a formula that would be easy to calculate in my head.

Good evening, I am pursuing a forestry degree right now and one of my labs we practiced inventorying a stand using a fixed radius plot. For the technical report we have to make a table of our recorded data but I am stuck on how to calculate the CF/A ;cubic feet per acre. I’ve tried searching every where on how to calculate it. Any help would be greatly appreciated!

Edit: 1/20 acre plot

Like potential energy is mgh, a linear function, movement energy is 0.5mv^2, a squared function.

Is there a function for energy that grows faster? Maybe a cubic or an exponential one?

My 6yo son has autism and is obsessed with watching number blocks videos on YouTube, memorizing their values, and asking me to make/print charts thereof before finally building them in Minecraft.

For the last month or two, he's loved "empty cubes." This video shows an example of "empty cube 9" whose cubic surface area construction requires 386 blocks based on the formula x = (n^3)-((n-2)^3) where n is the length of any side and x is the number of cubes requires to construct the total "empty cube" structure.

https://www.youtube.com/watch?v=pVd_Q6BLSns

He's now moved on to an obsession with hypercubes and has asked me to make a chart for him in Excel to print, but he has now officially exceeded my modest mathematical abilities. My assumption was that x = (n^4)-((n-2)^4) would be correct, which yields a pattern where the second and third values are 16 and 80. However, he is adamant it should be 16 followed by 74, but he cannot calculate any further. He is not verbal enough to explain to me where he's coming up with these figures.

Would anyone here be able to shed light on where I'm going wrong and perhaps provide a proper formula? He would be elated if I could make this happen for him somehow, and Google has not led me to the answer thus far.

Thanks in advance.

https://youtu.be/HWp8LFOZlPs

This is truly the best song I've ever heard. Take a look! I've watched it so many times, and I still don't remember the cubic formula.

I just want to see how insane it is :)

Long story short I fucked up and I have a math project tomorrow. I can't ask others to help me with the problem, so I won't post it, but I'm stuck. If I had a generalized formula for the line of symmetry of a cubic that would really really help. By that I mean just like how for quadratic the line of symmetry is -b/2a

You know the Quadratic’s formula . You can solve some tricky algebra question . In grade 12 in Vietnam , we know how to solve some “ complex “ equation , whose the “ x “ in a “ complex “ root too . But , how about Cubic’s formula ?

Yea , Cardano showed it before . But , sometimes , you decide yourself that the max power the x has means that how many answer this equation has . And I had read the formula in Wikipedia , but it seems can’t be understood for me .

Dear Grant, I've seen your videos for a long time now and always love when a new one comes out. I have a video idea. I started to enjoy math after graduating high school and hope to become a teacher one day, and I have learned a lot since graduating, either from YouTube videos like yours, or just doing stuff on my own.

There is one thing that I want to learn that I haven't yet. I've always been curious about how Gerolamo Cardano created the cubic formula. I've tried myself, but I haven't gotten far. Do you think you could do a video on the cubic equation and the cubic formula?

In this video (about 24:28) the speaker generates the single real solution to x^(3) - 6x - 40 = 0 using the cubic formula. The solution is x=4, but the cubic formula returns the expression

x = (20 - √392)^(1/3) + (20 + √392)^(1/3)

He does not explain how one might simplify this expression to give 4. Can anyone explain this particular result? Also, are there general methods for simplifying

(A - √B)^(1/3) + (A + √B)^(1/3)

where A and B are positive integers?

Thanks much.

I've posed myself with a challenge: to derive a solution to the general cubic equation without looking up the answer (I do not currently know the proper method for doing this). The way I like to attack the problem is by letting my geometric intuition guide me.

For the quadratic, I can simply "complete the square" by splitting the x term in half like equal rectangles placed on two sides of the same corner of the square (x^2). The missing square in the corner can be added to both sides of the equation and then x can be solved for in a straightforward way.

When I've glanced over solutions for the cubic, it looks like a bunch of seemingly random tricks that have no intuition guiding them. This is frustrating! I'm trying to let my intuition guide me by "completing the cube" among other things, but it seems like it might not work as nicely in the algebra.

Is this a fruitless endeavor? It's hard to accept that solving this problem boils down to a few tricks that just happen to work.

So I was recently trying to find sin(1°), and I found sin(3°) by angle subtraction formula [sin(18°-15°)], but in order to get to sin(1°) I need to solve a cubic polynomial

-4sin^3 (1)+3sin(1)-sin(3)=0

In terms of sin(1).

So far, I understand how to reduce a standard cubic into a depressed cubic through a Vieta substitution, but still don’t understand the rest of the process (that is, how those nested square roots got into the equation).

The best source I've found has been this one from UC Davis.

However, I don't understand how in step 3 he/she went from u^3 + v^3 = -q

to

u^3 * v^3 = -p^3 /27.

What is the cubic formula(s) for a cubic function that cannot be factored?