If you thought the Quadratic Formula was complicated, the method for solving Cubic Equations is
even more complex. We will use the example from the Cubic Equation Calculator:
2x^{3}  4x^{2}  22x + 24 = 0
Cubic equations have to be solved in several steps. First we define a variable 'f':
f = (3c/a)  (b²/a²)
3
"Plugging in" the numbers from the above equation, we get:
f = ((3 * 22/2)  (16/4)) / 3 =  12.333333...
Next we define 'g':
g = (2b³/a³)  (9bc/a²) + (27d/a)
27
From this point on, you are expected to "plug in" the numbers:
g = 4.07407407407407....
Then we define 'h':
h = (g²/4) + (f³/27)
h = 65.333333...
If h > 0, there is only 1 real root and is solved by another method.
(SCROLL down for this method)
For the special case where f=0, g=0 and h = 0, all 3 roots are real and equal.
(SCROLL to the bottom for this method)
When h <= 0, as is the case here, all 3 roots are real and we proceed as follows:


ALL 3 Roots Are Real
i = ((g²/4)  h) ^{½}
i = 8.33563754151978...
j = (i) ^{1/3}
j = 2.0275875100994063...
NOTE: The following trigonometric calculations are in radians
k = arc cosine ( (g / 2i))
k = 1.817673356517739...
L = j * 1
L = 2.0275875100994...
M = cosine (K/3)
M = 0.8219949365268...
N = (Square Root of 3) * sine (K/3)
N = 0.9863939238321...
P = (b/3a) * 1
P = 0.6666666666666...
x_{1} = 2j * cosine(k/3) (b/3a)
x_{1} = 4
x_{2} = L * (M + N) + P
x_{2} = 3
x_{3} = L * (M  N) + P
x_{3} = 1
When Only 1 Root Is Real
3x^{3}  10x^{2} + 14x + 27 = 0
f = (3c/a)  (b²/a²)
3
f = .962962962962962...
g = (2b³/a³)  (9bc/a²) + (27d/a)
27
g = 11.441700960219478...
h = (g²/4) + (f ³/27)
h = 32.761202560585275...
R = (g/2) + (h)^{½}
R = .002889779596782...
S = (R)^{1/3}
S = .142436591824886...
T = (g/2)  (h)^{½}
T = 11.4445907398163...
U = (T)^{1/3}
U = 2.25354770293599...
X_{1} = (S + U)  (b/3a)
X_{1} = 1
X_{2} = (S + U)/2  (b/3a) + i*(SU)*(3)^{½}/2
X_{2} = 2.16666666666... + i*2.07498326633146
X_{3} = (S + U)/2  (b/3a)  i*(SU)*(3)^{½}/2
X_{3} = 2.16666666666...  i*2.07498326633146
When All 3 Roots Are Real and Equal
x^{3} + 6x^{2} + 12x + 8 = 0
f = (3c/a)  (b²/a²)
3
f = ((3*12/1)(36/1)) / 3
f = 0
g = (2b³/a³)  (9bc/a²) + (27d/a)
27
g = ((2*216/1)  (9*6*12/1) + (27*8/1)) / 27
g = (432  648 + 216) / 27
g = 0
h = (g²/4) + (f³/27)
h=0
x_{1} = x_{2} = x_{3}= (d/a)^{1/3} * 1
x_{1} = x_{2} = x_{3}= (8/1)^{1/3} * 1
x_{1} = x_{2} = x_{3}= 2
RETURN TO CUBIC EQUATION CALCULATOR
