Please note that there may be other methods apart from this large formulas to solve cubics and quartics. But I wanted to show here that the formulas do exist.
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.
josechu2004@gmail.com