Solve the following cubic equation:
2x3 - 4x2 - 22x + 24 = 0
Calculate the discriminant Δ:
Δ = 4b3d - b2c2 + 4ac3 - 18abcd + 27a2d2
Δ = (4)-43(24) - -42-222 + (4)(2)-223 - 18(2)(-4)(-22)(24) + (27)22242
Δ = -6144 - 7744 + -85184 - 76032 + 62208
Δ = -112896
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)(-22)/2 - (-42)/(22) |
| 3 |
| f = | -33 - 4 |
| 3 |
| f = | -37 |
| 3 |
f = -12.333333333333 <------- Keep this in mind for later
Calculate g
| g = | 2b3/a3 - 9bc/a2 + 27d/a |
| 27 |
| g = | (2)(-43)/23 - (9)(-4)(-22)/22 + (27)(24)/2 |
| 27 |
| g = | (2)(-64)/8 - (9)(-4)(-22)/4 + (27)(24)/2 |
| 27 |
| g = | -16 - 198 + 324 |
| 27 |
| g = | 110 |
| 27 |
g = 4.0740740740741 <------- Keep this in mind for later
Calculate h
| h = | g2 |
| 4 |
| + |
| f3 |
| 27 |
| h = | 4.07407407407412 |
| 4 |
| + |
| -12.3333333333333 |
| 27 |
| h = | 16.598079561043 |
| 4 |
| + |
| -1876.037037037 |
| 27 |
h = 4.1495198902606 + -69.482853223594
h = -65.333333333333
Calculate i
i = √¼g2 - h
i = √¼(4.07407407407412) - -65.333333333333
i = √4.1495198902606 - -65.333333333333
i = √69.482853223594
i = 8.3356375415198
Calculate j
j = i(1/3)
j = 8.3356375415198(1/3)
j = 2.0275875100994
Calculate k
k = Arccosine(-g/(2i))
k = Arccosine(-4.0740740740741/((2)8.3356375415198))
k = Arccosine(--0.24437687302148))
k = 1.8176733565177
Calculate l
l = -j
l = -2.0275875100994)
l = -2.0275875100994
Calculate m
m = Cosine(k/3)
m = Cosine(1.8176733565177/3)
m = Cosine(0.60589111883925)
m = 0.82199493652679
Calculate n
n = 3 * Sin(k/3)
n = 3 * Sin(1.8176733565177/3)
n = 1.7320508075689 * Sin(0.60589111883925)
n = 1.7320508075689 * 0.5694947974515
n = 0.98639392383214
Calculate p
| p = | -b |
| 3a |
| p = | --4 |
| 3(2) |
| p = | 4 |
| 6 |
p = 0.66666666666667
Calculate the first root x1
x1 = 2j * Cosine(k/3) - b/(3a)
x1 = (2)(2.0275875100994) * (Cosine(1.8176733565177/3) - -4/(3)(2)
x1 = (4.0551750201988)(Cosine(0.60589111883925) - -0.66666666666667
x1 = 3.3333333333333 - -0.66666666666667
x1 = 4
Calculate the second root x2:
x2 = l(m + n) + p
x2 = -2.0275875100994(0.82199493652679 + 0.98639392383214) + 0.66666666666667
x2 = -2.0275875100994(1.8083888603589) + 0.66666666666667
x2 = -3.6666666666667 + 0.66666666666667
x2 = -3
Calculate the third root x3
x3 = l(m - n) + p
x3 = -2.0275875100994(0.82199493652679 - 0.98639392383214) + 0.66666666666667
x3 = -2.0275875100994(-0.16439898730536) + 0.66666666666667
x3 = 0.33333333333333 + 0.66666666666667
x3 = 1
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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