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 $latex f(x) = ax^3 + bx^2 + cx + d$, 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 $latex ax^4 + bx^3 + cx^2 + dx + e=0$, 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: