How to solve the cubic equation for quartic critical points

[1plus1is10]

I need this solved for x:
y' = 4ax^3 + 3bx^2 + 2cx + d = 0

This is to say, I need the formula for the "critical points" of a Quartic function.
Wikipedia says: "The derivative of a quartic function is a cubic function."
https://en.wikipedia.org/wiki/Quartic_function

And I found the above derivative here:
http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Glazer/essays/HTML/quartinfsola.html

Any help solving it would be great (it is beyond my ability).
Thanks

 

[fresh_42]

It's a bit tricky, but here's how it goes:
https://en.wikipedia.org/wiki/Cubic..._to_the_cubic_equation_with_real_coefficients

 

[1plus1is10]

No kidding. Tricky is definitely the word for it.

I've previously read and re-read that whole page already, and unfortunately, I'm clueless with any high-level math. So it would be impossible for me to use the general cubic equation as an example for me to do the same math tricks to the above cubic.

I typically Google until I find what I need. Meaning, I need a website to say "this is the formula for XYZ" (and the formula need to be in a form that does not have many Greek letters). Otherwise, I keep searching. It all comes down to: Can I convert it to computer code?

Therefore, any spoon-feeding would be greatly appreciated.

 

[fresh_42]

If you have no numbers for $a,b,c,d$ then you're stuck with the giant formula I quoted. There is no easy way in this generality.

 

[1plus1is10]

Looking closer at it. That's just crazy.
How can a simple derivative explode into something that big?

I'm just going to "graph" my Quartic's into a data set and then search for highs and lows instead.

Sorry to waste your time.
Thanks anyway.