Representing the Quartic Formula

I have used various graphics packages to hunt for a quartic formula. Over the years I had settled for a formula that is one order down, the cubic formula. For a polynomial of the form f(x) = ax^3 + bx^2 + cx + d, all real solutions may be obtained by the formulae:

These formulae are relatively easy to find with a math package like Maple. See how the formula is quite a lot more complex when compared with the quadratic formula.

The formula for finding all real solutions to an order 4 polynomial has been elusive, however. Maple simply gives up and doesn’t bother. There is, however, Mathematica, which can come up with a quartic formula. I have a screenshot of all four solutions to ax^4 + bx^3 + cx^2 + dx + e=0, two parts at a time from the Mathematica output: (to view, you need to right click on the image and select “View image” or something similar on your browser)

The second part of this is:

Thank God Neils Abel verified that there is no such formula polynomials of order 5 and above. Another blogger went to the trouble of writing all four formulae in LaTeX, and came up with:

 

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I am Paul King, a math and science teacher. I help maintain the MIT FAQ Archive along with Nick Bolach. I am also the maintainer of the FAQ for sci.bio.food-science.