3x^4 + 6x^3 + -123x^2 + -126x + e = 0

Enter quartic equation coefficients:

x⁴

x³

x²

= 0

Solve the quartic equation
3x⁴ + 6x³ - 123x² - 126x + 1080 = 0

Divide the quartic by x⁴ coefficient of 3
Our resulting equation is
x⁴ + 2x³ - 41x² - 42x + 360 = 0
where our new (a,b,c,d,e) coefficients become (1,2,-41,-42,360)

Calculate preliminary values

f = c - (⅜b²)
f = -41 - ⅜(2)²
f = -41 - ⅜(4)
f = -41 + 1.5
f = -42.5

g = d + ⅜b³ - ½bc
g = -42 + ⅜(2³) - ½(2 * -41)
g = -42 + ⅜(8) - ½(-82)
g = -42 + 1 + 41
g = 0

h = e - (3b⁴/256) + (b² * c)/16 - bd/4
h = 360 - (3(2⁴)/256) + (2² * -41/16) - (2 * -42)/4
h = 360 - (3(16)/256) - 10.25 + 84/4
h = 360 - 0.1875 - 10.25 + 21
h = 370.5625

Form a cubic equation:

x³ + ½fx² + ((f² - 4h)/16)x - g²/64 = 0

Find the a, b, c, and d coefficients

Calculate the b coefficient:

b = ½f
b = ½(-42.5)
  b  =  -21.25

Calculate the c coefficient:

c  =	f2 - 4h
	16

c  =	-42.52 - 4(370.5625)
	16

c  =	1806.25 - 1482.25
	16

c  =	324
	16

  c  =  20.25

Calculate the d coefficient:

d  =	-g2
	64

d  =	-02
	64

d  =	0
	64

  d  =  0

Solve the following cubic equation
x³ - 21.25x² + 20.25x + 0 = 0

Calculate the discriminant Δ:

Δ = 4b³d - b²c² + 4ac³ - 18abcd + 27a²d²
Δ = (4)-21.25³(0) - -21.25²20.25² + (4)(1)20.25³ - 18(1)(-21.25)(20.25)(0) + (27)1²0²
Δ = -0 - 185168.84765625 + 33215.0625 - -0 + 0
Δ = -151953.78515625

Since Δ < 0:
The cubic equation has 3 real roots.

Calculate f:

f  =	(3c/a) - b2/a2
	3

f  =	(3)(20.25)/1 - (-21.252)/(12)
	3

f  =	60.75 - 451.5625
	3

f  =	-390.8125
	3

  f  =  -130.27083333333

Calculate g:

g  =	2b3/a3 - 9bc2 + 27d/a
	27

g  =	[(2)(-21.25313 - (9)(-21.25)20.252 + (27)(0)/1
	27

g  =	[(2)(-21.25313 - (9)(-21.25)20.252 + (27)(0)/1
	27

g  =	-19191.40625 - -3872.8125 + 0
	27

g  =	-15318.59375
	27

  g  =  -567.35532407407

Calculate h:

h  =	g2
	4

+

f3

27

h  =	-567.355324074072
	4

+

-130.270833333333

27

h  =	321892.0637552
	4

+

-2210759.8766366

27

  h  =  80473.015938799 + -81879.995430987

  h  =  -1406.9794921875

Calculate i:

  i  =  Square Root(¼g² - h)

  i  =  Square Root(¼(-567.35532407407²) - -1406.9794921875)

  i  =  √80473.015938799 - -1406.9794921875

  i  =  √81879.995430987

  i  =  286.14680747998

Calculate j:

  j  =  i^(1/3)

  j  =  286.14680747998^(1/3)

  j  =  6.5896594078231

Calculate k:

  k  =  Arccosine(-g/(2i))

  k  =  Arccosine(--567.35532407407/((2)286.14680747998))

  k  =  Arccosine(0.99137105367454)

  k  =  0.13146394750086

Calculate l:

  l  =  -j

  l  =  -(6.5896594078231)

  l  =  -6.5896594078231

Calculate m:

  m  =  Cosine(k/3)

  m  =  Cosine(0.13146394750086/3)

  m  =  Cosine(0.04382131583362)

  m  =  0.99903999977871

Calculate n:

  n  =  √3 * Sin(k/3)

  n  =  √3 * Sin(0.13146394750086/3)

  n  =  1.7320508075689 * Sin(0.04382131583362)

  n  =  1.7320508075689 * 0.043807292111654

  n  =  0.075876455679396

Calculate p:

p  =	-b
	3a

p  =	-(-21.25)
	3(1)

p  =	21.25
	3

  p  =  7.0833333333333

Calculate the first root x₁:

  x₁  =  2j * Cosine(k/3) - b/(3a)

  x₁  =  (2)(6.5896594078231) * (Cosine(0.13146394750086/3) - -21.25/(3)(1)

  x₁  =  (13.179318815646)(Cosine(0.04382131583362) - -7.0833333333333

  x₁  =  13.166666666667 - -7.0833333333333

  x₁  =  20.25

Calculate the second root x₂:

  x₂  =  l(m + n) + p

  x₂  =  -6.5896594078231(0.99903999977871 + 0.075876455679396) + 7.0833333333333

  x₂  =  -6.5896594078231(1.0749164554581) + 7.0833333333333

  x₂  =  -7.0833333333333 + 7.0833333333333

  x₂  =  -0

Calculate the third root x₃:

  x₃  =  l(m - n) + p

  x₃  =  -6.5896594078231(0.99903999977871 - 0.075876455679396) + 7.0833333333333

  x₃  =  -6.5896594078231(0.92316354409931) + 7.0833333333333

  x₃  =  -6.0833333333333 + 7.0833333333333

  x₃  =  1

You have found all 3 Real roots now.
(20.25, -0, 1)

Continue solving the quartic:

Calculate p and q

p = √20.25
p = 4.5

q = √1
q = 1

Calculate the r coefficient:

r  =	-g
	8pq

r  =	-(0)
	8(4.5)(1)

r  =	0
	36

  r  =  0

Calculate the s coefficient:

s  =	b
	4a

s  =	2
	4(1)

s  =	2
	4

  s  =  0.5

Now we have everything we need for our quartic equation roots:

x₁ = p + q + r - s
x₁ = 4.5 + 1 + 0 - 0.5
x₁ = 5

x₂ = p - q - r - s
x₂ = 4.5 - 1 - 0 - 0.5
x₂ = 3

x₃ = -p + q - r - s
x₃ = -4.5 + 1 - 0 - 0.5
x₃ = -4

x₄ = -p - q + r - s
x₄ = -4.5 - 1 + 0 - 0.5
x₄ = -6

---

x₁ = 5
x₂ = 3
x₃ = -4
x₄ = -6