Solve the following cubic equation:
3x3 - 6x2 - 99x + 270 = 0
Calculate the discriminant Δ:
Δ = 4b3d - b2c2 + 4ac3 - 18abcd + 27a2d2
Δ = (4)-63(270) - -62-992 + (4)(3)-993 - 18(3)(-6)(-99)(270) + (27)322702
Δ = -233280 - 352836 + -11643588 - 8660520 + 17714700
Δ = -3175524
Since Δ < 0, our cubic equation has 3 real roots.
A few interim calculations need to be done to get our 3 answers.
Calculate f
| f = | (3c/a) - b2/a2 |
| 3 |
| f = | (3)(-99)/3 - (-62)/(32) |
| 3 |
| f = | -99 - 4 |
| 3 |
| f = | -103 |
| 3 |
f = -34.333333333333 <------- Keep this in mind for later
Calculate g
| g = | 2b3/a3 - 9bc/a2 + 27d/a |
| 27 |
| g = | (2)(-63)/33 - (9)(-6)(-99)/32 + (27)(270)/3 |
| 27 |
| g = | (2)(-216)/27 - (9)(-6)(-99)/9 + (27)(270)/3 |
| 27 |
| g = | -16 - 594 + 2430 |
| 27 |
| g = | 1820 |
| 27 |
g = 67.407407407407 <------- Keep this in mind for later
Calculate h
| h = | g2 |
| 4 |
| + |
| f3 |
| 27 |
| h = | 67.4074074074072 |
| 4 |
| + |
| -34.3333333333333 |
| 27 |
| h = | 4543.7585733882 |
| 4 |
| + |
| -40471.37037037 |
| 27 |
h = 1135.9396433471 + -1498.9396433471
h = -363
Calculate i
i = √¼g2 - h
i = √¼(67.4074074074072) - -363
i = √1135.9396433471 - -363
i = √1498.9396433471
i = 38.716141896463
Calculate j
j = i(1/3)
j = 38.716141896463(1/3)
j = 3.3829638550307
Calculate k
k = Arccosine(-g/(2i))
k = Arccosine(-67.407407407407/((2)38.716141896463))
k = Arccosine(--0.87053363410632))
k = 2.6270819915826
Calculate l
l = -j
l = -3.3829638550307)
l = -3.3829638550307
Calculate m
m = Cosine(k/3)
m = Cosine(2.6270819915826/3)
m = Cosine(0.87569399719418)
m = 0.64046403080679
Calculate n
n = 3 * Sin(k/3)
n = 3 * Sin(2.6270819915826/3)
n = 1.7320508075689 * Sin(0.87569399719418)
n = 1.7320508075689 * 0.76798816738458
n = 1.3301945255218
Calculate p
| p = | -b |
| 3a |
| p = | --6 |
| 3(3) |
| p = | 6 |
| 9 |
p = 0.66666666666667
Calculate the first root x1
x1 = 2j * Cosine(k/3) - b/(3a)
x1 = (2)(3.3829638550307) * (Cosine(2.6270819915826/3) - -6/(3)(3)
x1 = (6.7659277100615)(Cosine(0.87569399719418) - -0.66666666666667
x1 = 4.3333333333333 - -0.66666666666667
x1 = 5
Calculate the second root x2:
x2 = l(m + n) + p
x2 = -3.3829638550307(0.64046403080679 + 1.3301945255218) + 0.66666666666667
x2 = -3.3829638550307(1.9706585563286) + 0.66666666666667
x2 = -6.6666666666667 + 0.66666666666667
x2 = -6
Calculate the third root x3
x3 = l(m - n) + p
x3 = -3.3829638550307(0.64046403080679 - 1.3301945255218) + 0.66666666666667
x3 = -3.3829638550307(-0.689730494715) + 0.66666666666667
x3 = 2.3333333333333 + 0.66666666666667
x3 = 3
Final Answer
You have 2 free calculationss remaining
What is the Answer?
How does the Cubic Equation Calculator work?
1) Solve the long way for all 3 roots and the discriminant Δ
2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable.
This calculator has 5 inputs.
What 4 formulas are used for the Cubic Equation Calculator?
x1 = 2j * Cosine(k/3) - b/(3a)
x2 = l(m + n) + p
x3 = l(m - n) + p
For more math formulas, check out our Formula Dossier
What 7 concepts are covered in the Cubic Equation Calculator?
- cubic
- cubic equation
- An equation of the form ax3 + bx2 + cx + d = 0
- equation
- a statement declaring two mathematical expressions are equal
- quadratic
- Polynomials with a maximum term degree as the second degree
- rational root theorem
- used to find the rational solutions of a polynomial equation
- synthetic division
- a shorthand method for dividing a polynomial by a linear factor
- unknown
- a number or value we do not know
Example calculations for the Cubic Equation Calculator
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