From: DBRS28A@prodigy.com (Mr. Gerald A. Sabin)
Newsgroups: sci.math
Subject: General solution to quartic (PLEASE)
Date: 27 Jul 1998 12:27:32 GMT

Solution to Quartic,
Method I
(known as the Ferrari method)
The general quartic is,
  A4x^4 + A3x^3 + A2x^2 + A1x + A0 = 0  Eqn 1
where the Ai are real coefficients.
Divide thru by A4 and get
  x^4 + B3x^3 + B2x^2 + B1x + B0 = 0  Eqn 2
complete the square by adding
  (Rx + S)^2 to both sides
  (x^2 + (B3/2)x + T/2)^2 = (Rx + S)^2  Eqn 3

R,S,T are constants yet to be determined.

Collect coefficients of powers of x and equate them
on both Eqn 2 and Eqn 3.
  R^2 = T + (B3^2)/4 - B2  Eqn 4
  RS  = B3*T/4 - B1/2  Eqn 5
  S^2 = T^2/4 - B0  Eqn 6
Multiply Eqn 4 by Eqn 6 and equate result to (Eqn 5)^2
This yields an equation for T:
  T^3 - B2*T^2 +(B3*B1-4*B0)T - B1^2 - B3^2*B0 = 0 Eqn 7

At this point, the QUARTIC has been transformed into a
CUBIC that needs to be solved.
All of the above, plus the subsequent solution of the
CUBIC can be written as a program in QBASIC.

After that, you can solve QUARTICS all day long.

Method II
This is a Numerical Method that involves successive
approximation.
It is presented in Handbook of Mathematical Functions
 (AMS 55) by Abramowitz & Stegun
  Equation: X^4 + A*X^3 + B*X^2 + C*X + D = 0
  we seek the quadratic factors
  (X^2 + P1*X + Q1) * (X^2 + P2*X + Q2)

Assume a value for Q1, and divide it into D
Make a computation to determine how to adjust Q1
and try again.
..etc...
This method also is readily programmable by QBASIC.

Best wishes,
  -Jerry
   email: Jerry_Sabin.NOSPAM@prodigy.com
          (remove .NOSPAM)