Cubic Equation Calculator

Cubic Equation Calculator 

The cubic equation calculator is an online maths tool to find the roots of a cubic equation. While finding the real, imaginary, or both roots, it provides complete calculations.

Once you know how to find the roots of the cubic equations by hand later in the article, you will be grateful for this tool.

What is a cubic equation?

The syntax of the cubic equation is 

ax³ + bx² + cx + d.

Where a,b,c, and d are called its coefficients. 

Cube: 

A cube is a 3-dimensional figure. It is similar to a square but different at the same time. It is measured in cubic feet and cubic meters.

The polynomial form of cube and square differ as well. A square has a second-degree polynomial, the roots of which are calculated by the quadratic formula. While the cube has a third-degree polynomial. 

Roots:

Since a cubic polynomial has 3 as the highest power, it will have 3 roots. These roots must have one real root. The other two can be real or a pair of conjugates.

Cubic equation formula:

The formula for the calculation of the roots of the cubic equation is very complex. Each argument X, 3 in the cubic polynomial, has its own formula. 

X₁ = S + T - b/3a 
X₂ = -(S + T)/2 - (S - T) * i√3/2 - b/3a
X₃ = -(S + T)/2 + (S - T) * i√3/2 - b/3a

It is called the Cardano formula. The variable S and T are still unknown in the above formulas. Their formulas are:

S = ∛(R + √(Q³ + R²))
T=∛(R - √(Q³ + R²))

Again, the variables Q and R are calculated by different formulas. 

Q = (3ac - b²) / (9a²)
R = (9abc - 27a²d - 2b³) / (54a³)

How to find the roots of a cubic equation?

Start by calculating the values in a backward manner. Here is a guide.

1. Make the cubic equation a standard equation. 
2. Find the values of Q and R by using the values of variables of the equation. 
3. Use the Q and R to find S and T and after that X₁, X₂, and X₃.

Look at the following example using the Cardano formula.

Example:

What are the roots of the cubic polynomial 2x³ + 3x² + 4x + 1. 

Solution:

Step 1: Make an equation and separate the variables.

2x³ + 3x² + 4x + 1 = 0

a = 2
b = 3
c = 4
d = 1

Step 2: Find Q and R.

Finding Q:

Q = (3ac - b²) / (9a²)
Q = {(3)(2)(4) - (3)²} / {9(2)²}
Q = (24 - 9) / (9*4)
Q = 0.4167

Finding R:

R = (9abc - 27a²d - 2b³) / (54a³)
R = [{(9)(2)(3)(4)} - {(27)(2)2(1)} - {(2)(3)3}] / {54(2)³}
R = (216) - (108) - (54) / 432
R = 54 / 432
R = 0.1250
 
Step 3: Fins S and T.

Finding S:

S = ∛(R + √(Q³ + R²))
S = ∛[0.1250 + √{(0.4167)³ + (0.1250)²}]
S = ∛[0.1250 + √{(0.0724) + (0.0156)}]
S = ∛[0.1250 + √(0.0880)]
S =  ∛[0.1250 + 0.2966]
S = 0.7498

Finding T:

T=∛(R - √(Q³ + R²))
T = ∛[0.1250 - √{(0.4167)³ + (0.1250)²}]
T = ∛[0.1250 - √{(0.0724) + (0.0156)}]
T = ∛[0.1250 - √(0.0880)]
T = ∛[0.1250 - 0.2966]
T = -0.5557

Step 4: Find the roots.

First root (X₁):

X₁ = S + T - b/3a 
     = 0.7498 + -0.5557 - 3/(3)(2)
     = 0.7498 - 0.5557 - 3/6
     = -0.3059

Second root (X₂):

X₂ = -(S + T)/2 - (S - T) * i√3/2 - b/3a
     = -((0.7498) + (-0.5557))/2 + ((0.7498) - (-0.5557))* i√3/2 - (3)/(3)(2)
     = -(0.7498-0.5557)/2 + (1.3054999999)* (0.867)i - (3)/(6)
     = -(0.0971) - (0.5000) + (1.3054999999) * (0.867)i
     = -0.5971 + 1.1319i

Third root (X₃):

X₃ = -(S + T)/2 + (S - T) * i√3/2 - b/3a
   = -((0.7498) + (-0.5557))/2 - ((0.7498) - (-0.5557))* i√3/2 - (3)/(3)(2)
   = -(0.7498-0.5557)/2 - (1.3054999999)* (0.867)i - (3)/(6)
   = -(0.0971) - (0.5000) - (1.3054999999) * (0.867)i
   = -0.5971 - 1.1319i

The roots are -0.3059, -0.5971 + 1.1319i, and -0.5971 - 1.1319i