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If you thought the Quadratic Formula was complicated, the method for solving Cubic Equations is
even more complex.
Let's use the equation from the Cubic Equation Calculator as our first example:
2x3 - 4x2 - 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:
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ALL 3 Roots Are Real
We just calculated the values of 'f','g' and 'h' so let's calculate the rest.
i = ((g²/4) - h)½
i = 8.33563754151978...
j = (i) ⅓
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...
x1 = 2j • cosine(k/3) -(b/3a)
x1 = 4
x2 = L • (M + N) + P
x2 = -3
x3 = L • (M - N) + P
x3 = 1
When Only 1 Root Is Real
3x3 - 10x2 + 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) + (f³/27)
h = 32.761202560585275...
R = -(g/2) + (h)½
R = .002889779596782...
S = (R)⅓
S = .142436591824886...
T = -(g/2) - (h)½
T = -11.4445907398163...
U = (T)⅓
U = -2.25354770293599...
X1 = (S + U) - (b/3a)
X1 = -1
X2 = -(S + U)/2 - (b/3a) + i•(S-U)•(3)½/2
X2 = 2.16666666666... + i•2.07498326633146
X3 = -(S + U)/2 - (b/3a) - i•(S-U)•(3)½/2
X3 = 2.16666666666... - i•2.07498326633146
When All 3 Roots Are Real and Equal
x3 + 6x2 + 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
x1 = x2 = x3= (d/a)1/3 • -1
x1 = x2 = x3= (8/1)1/3 • -1
x1 = x2 = x3= -2
RETURN TO CUBIC EQUATION CALCULATOR
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