Solve the following cubic equation:
3x3 + 21x2 - 18x - 216 = 0
Calculate the discriminant Δ:
Δ = 4b3d - b2c2 + 4ac3 - 18abcd + 27a2d2
Δ = (4)213(-216) - 212-182 + (4)(3)-183 - 18(3)(21)(-18)(-216) + (27)32-2162
Δ = -8001504 - 142884 + -69984 - 4408992 + 11337408
Δ = -1285956
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)(-18)/3 - (212)/(32) |
| 3 |
| f = | -18 - 49 |
| 3 |
| f = | -67 |
| 3 |
f = -22.333333333333 <------- Keep this in mind for later
Calculate g
| g = | 2b3/a3 - 9bc/a2 + 27d/a |
| 27 |
| g = | (2)(213)/33 - (9)(21)(-18)/32 + (27)(-216)/3 |
| 27 |
| g = | (2)(9261)/27 - (9)(21)(-18)/9 + (27)(-216)/3 |
| 27 |
| g = | 686 - -378 + -1944 |
| 27 |
| g = | -880 |
| 27 |
g = -32.592592592593 <------- Keep this in mind for later
Calculate h
| h = | g2 |
| 4 |
| + |
| f3 |
| 27 |
| h = | -32.5925925925932 |
| 4 |
| + |
| -22.3333333333333 |
| 27 |
| h = | 1062.2770919067 |
| 4 |
| + |
| -11139.37037037 |
| 27 |
h = 265.56927297668 + -412.56927297668
h = -147
Calculate i
i = √¼g2 - h
i = √¼(-32.5925925925932) - -147
i = √265.56927297668 - -147
i = √412.56927297668
i = 20.311801322795
Calculate j
j = i(1/3)
j = 20.311801322795(1/3)
j = 2.7284509239575
Calculate k
k = Arccosine(-g/(2i))
k = Arccosine(--32.592592592593/((2)20.311801322795))
k = Arccosine(-0.80230679875783))
k = 0.63964652947629
Calculate l
l = -j
l = -2.7284509239575)
l = -2.7284509239575
Calculate m
m = Cosine(k/3)
m = Cosine(0.63964652947629/3)
m = Cosine(0.21321550982543)
m = 0.97735555485044
Calculate n
n = 3 * Sin(k/3)
n = 3 * Sin(0.63964652947629/3)
n = 1.7320508075689 * Sin(0.21321550982543)
n = 1.7320508075689 * 0.21160368475758
n = 0.36650833306892
Calculate p
| p = | -b |
| 3a |
| p = | -21 |
| 3(3) |
| p = | -21 |
| 9 |
p = -2.3333333333333
Calculate the first root x1
x1 = 2j * Cosine(k/3) - b/(3a)
x1 = (2)(2.7284509239575) * (Cosine(0.63964652947629/3) - 21/(3)(3)
x1 = (5.456901847915)(Cosine(0.21321550982543) - 2.3333333333333
x1 = 5.3333333333333 - 2.3333333333333
x1 = 3
Calculate the second root x2:
x2 = l(m + n) + p
x2 = -2.7284509239575(0.97735555485044 + 0.36650833306892) + -2.3333333333333
x2 = -2.7284509239575(1.3438638879194) + -2.3333333333333
x2 = -3.6666666666667 + -2.3333333333333
x2 = -6
Calculate the third root x3
x3 = l(m - n) + p
x3 = -2.7284509239575(0.97735555485044 - 0.36650833306892) + -2.3333333333333
x3 = -2.7284509239575(0.61084722178153) + -2.3333333333333
x3 = -1.6666666666667 + -2.3333333333333
x3 = -4
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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