IMBROGLIO ABSCISSATRON 1.0 - THE QUARTIC FORMULA
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Below is the most commonly used formula for solving the
general quartic (fourth-order) equation.

1. Original quartic equation in general form (the x^4 term must have a coefficient of 1):
	x^4 + ax^3 + bx^2 + cx + d

2. Let x = y - a/4, this gives:
	x^4 + ax^3 + bx^2 + cx + d =

	= (y - a/4)^4 + a(y - a/4)^3 + b(y - a/4)^2 + c(y - a/4) + d =

	= y^4 - ay^3 + 3a^2y^2/8 - ya^3/16 + a^4/256 +
          + ay^3 - 3y^2a^2/4 + 3ya^3/16 - a^4/64 +
          + by^2 - bay/2 + ba^2/16 + cy - ac/4 + d =

	= y^4 + y^3(- a + a) + y^2(3a^2/8 - 3a^2/4 + b) +
	  + y(-a^3/16 + 3a^3/16 - ba/2 + c) +
	  + (a^4/256 - a^4/64 + ba^2/16 - ac/4 + d) =

	= y^4 + qy^2 + ry + s
	  (the y^3 term is eliminated)

3. Calculate q, r, and s:
	q = 3a^2/8 - 3a^2/4 + b
	r = -a^3/16 + 3a^3/16 - ba/2 + c
	s = a^4/256 - a^4/64 + ba^2/16 - ac/4 + d

4. Factorise:
	y^4 + qy^2 + ry + s = (y^2 + ky + l)(y^2 - ky + m)
	(finding k, l, and m reduces the equation to two quadratics)

5. Equating like powers of y by expanding the right hand side gives equations for k, l, and m:
	l + m - k^2 = q
	k(m - l)    = r 
	lm          = s

	Therefore, m and l are obtained:
		m = (k^2 + q + r/k)/2
		l = (k^2 + q - r/k)/2

6. Substituting these into the third equation gives:
	(k^3 + qk + r)(k^3 + qk - r) = 4sk^2 

	Which becomes:
		k^6 + 2qk^4 + (q^2 - 4s)k^2 - r^2 = 0
		(which is a cubic equation for k^2 and can be solved using the cubic formula)

7. With a few backwards substitutions, the original value of x can be obtained.


Summary:

	1. Original equation in general form:
		x^4 + ax^3 + bx^2 + cx + d
	2. Calculate q, r, and s:
		q = 3a^2/8 - 3a^2/4 + b
		r = -a^3/16 + 3a^3/16 - ba/2 + c
		s = a^4/256 - a^4/64 + ba^2/16 - ac/4 + d
	3. Solve the cubic equation for one real root:
		z^3 + 2qz^2 + (q^2 - 4s)z - r^2 = 0
		k = sqrt(z)
	4. Calculate l and m:
		m = (k^2 + q + r/k)/2
		l = (k^2 + q - r/k)/2
	5. Replace in k, l, m and solve the quadratic equations:
		(y^2 + ky + l)(y^2 - ky + m) = 0
	6. Find the original unknown:
		x = y - a/4

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Copyright 2000 - Imbroglio Inc.