The general form of the 4th degree equation (or Quartic) is:  ax4 + bx3 + cx2 + dx + e = 0

Quartics have 4 roots.

The 4 roots can be represented this way:

First root (of four):

Second root (of four):

Third root (of four):

Fourth root (of four):

The four formulas are equal except for a "+ or -" sign at the beginning, and a couple of "+ or -" signs at the far right end of the images.  Note that the imaginary unit "i" does not appear in any of these formulas.


Now, the same four formulas in ASCII.  The differences are highlighted here in yellow:

x = -b/(4 a) - 1/2 Sqrt[b^2/(4 a^2) - 2 c/(3 a) + 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)] - 1/2 Sqrt[b^2/(2 a^2) - 4 c/(3 a) - 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) - 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3) - (-b^3/a^3 + 4 b c/a^2 - 8 d/a)/(4 Sqrt[b^2/(4 a^2) - 2 c/(3 a) + 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a ((2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2]))^(1/3)) + 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)])]

x = -b/(4 a) - 1/2 Sqrt[b^2/(4 a^2) - 2 c/(3 a) + 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)] + 1/2 Sqrt[b^2/(2 a^2) - 4 c/(3 a) - 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) - 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3) - (-b^3/a^3 + 4 b c/a^2 - 8 d/a)/(4 Sqrt[b^2/(4 a^2) - 2 c/(3 a) + 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a ((2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2]))^(1/3)) + 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)])]

x = -b/(4 a) + 1/2 Sqrt[b^2/(4 a^2) - 2 c/(3 a) + 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)] - 1/2 Sqrt[b^2/(2 a^2) - 4 c/(3 a) - 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) - 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3) + (-b^3/a^3 + 4 b c/a^2 - 8 d/a)/(4 Sqrt[b^2/(4 a^2) - 2 c/(3 a) + 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a ((2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2]))^(1/3)) + 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)])]

x = -b/(4 a) + 1/2 Sqrt[b^2/(4 a^2) - 2 c/(3 a) + 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)] + 1/2 Sqrt[b^2/(2 a^2) - 4 c/(3 a) - 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) - 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3) + (-b^3/a^3 + 4 b c/a^2 - 8 d/a)/(4 Sqrt[b^2/(4 a^2) - 2 c/(3 a) + 2^(1/3) (c^2 - 3 b d + 12 a e)/(3 a ((2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2]))^(1/3)) + 1/(3 2^(1/3) a)(2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)])]

The formulas on top were obtained with Wolfram's program Mathematica.  There seemed not to be an option, so I copied them by hand into ASCII.
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