Thread: General Solution to the Cubic question using the Wolfram site example

Hello.
I have been trying to follow the solution on this site:
Cubic Formula -- from Wolfram MathWorld

and have been making headway in understanding it. However, I am getting hung up around lines 35, 36, and 37. I cannot work out how they reduced and arrived at everything behind -QB+2R on line 36, nor how that reduces to -QB on line 37.

What am i missing?

Wow. This method is a complete mess. Cardano's solution is easier to follow, in my opinion.

Anyway, the lines you mention are the product B^3 = B^2 x B. What specifically about those lines is troubling you? I can't know until I see your work.

-Dan

I was wondering how this bit: [ R+(Q³ + R²)^1/2]^1/3[R-(Q³+R²)^1/2]^2/3 + [R+(Q³+R²)^1/2]^2/3[R-(Q³+R²)^1/2]^1/3

reduced to this bit:

[R²-(Q³+R²)]^1/3 * [(R+(Q³+R²)^1/2)^1/3 + (R-(Q³+R²)^1/2)^1/3]

On the way home last night, I realized that you can extract this as a factor from the first bit:

(R+(Q³+R²)^1/2)^1/3(R-(Q³+R²)^1/2)^1/3

To get what Wolfram was going for.
Things fell into place after that.

Believe it or not, that Wolfram page seemed to be the best help out of all of the pages regarding the Cubic that I found. Most pages attempting to explain the solution, seemed to skip over, or gloss over steps or reasonings. You have a better recommendation perhaps?

Originally Posted by zendo

Believe it or not, that Wolfram page seemed to be the best help out of all of the pages regarding the Cubic that I found. Most pages attempting to explain the solution, seemed to skip over, or gloss over steps or reasonings. You have a better recommendation perhaps?

I've only seen and investigated Cardano's method. It only gives the real solution to the cubic and you then have to find the other two by the quadratic formula. Yours gives all three solutions in the general form (not the trigonometric form) of the solution.

It just depends on what you need out of the solutions I guess. Anyway I did take a closer look at the Wolfram solution and I think I'm more in tune with the logic of the solution, but I still think it's a mess.

-Dan