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this is a follow up question to one I asked recently here: How to solve this quartic?

I have the equation $x^4$+4a$x^3$+(4$a^2$+1)$x^2$−1=0, whereas I'm trying to find the solutions of x in terms of a.

My previous question asked whether there was a way to solve this without using the quartic formula. There isn't as far as I know, but if you know of one by all means answer there.

Now I have been trying to solve this using the quartic formula. My problem is that I haven't been able to find any program online to do this with. Eventually, if I tried I could probably break it down for online calculators, but there's multiple kinks that keep coming up whenever I try to do this. Solving it by hand would take a while too (obviously).

Basically I'm just looking for a nice way to get the answer. I'ts a weird thing to be stuck on, since I know exactly what I need to do, I just don't have any good way to do it.

I'm at a loss here so all suggestions are welcome. Thank you.

Joey Peluka
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  • You will spare a step by solving $1+4ax^{-1}+(4a^2+1)x^{-2}−x^{-4}=0$ for $x^{-1}$, which is already in depressed form. But don't expect an easy formula. –  Mar 14 '20 at 20:25

2 Answers2

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My problem is that I haven't been able to find any program online to do this with.

Here is an answer using WolframAlpha.

For example, here's one solution it gives:

enter image description here

And if you follow the link, you'll find the other 3.

Jordan
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  • Why should a copy pasted image file of an equation generated automatically from a computer be worthy of praise? No originality or thought put into it. If you can find it on google of by having wolfram solve it for you then an external link to the source would have sufficed. Exactly the sort of linking you criticize my answer for. You cant even be intellectually self-consistent, can you? – CogitoErgoCogitoSum Mar 15 '20 at 14:48
  • Just FYI, stackexchange has a pre-existing template for answers that use WolframAlpha. As soon as I pasted the WolframAlpha link, it suggested the format for the post above.

    Furthermore, originality/creativity is not needed to answer this question. This was a technical question with a straightforward answer. Wolfram shows the steps, so it is quite useful in this scenario.

     – Jordan Apr 07 '20 at 07:04
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Well, instead of looking to the quartic formula directly, why not follow the math of its derivation. Its infinitely easier to "complete the square", is it were, than to plug and play with the quadratic formula. Okay, thats not true for quadratics, but it is true for the quartic equivalent of the statement.

See this question asked by a patron several years ago. My answer therein may prove particularly useful to you. Is there a general formula for solving 4th degree equations (quartic)?

I find that actually "solving" a quartic is much more straightforward and much easier to handle than trying to either use or memorize those ridiculous set of formulas.

CogitoErgoCogitoSum
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