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Cubic Equation Solver

To solve cubic equation input coefficients of a, b, c, & d and press calculate button using cubic equation solver

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ax³ + bx² + cx + d = 0
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  Find the roots of a third-degree polynomial with the help of a cubic equation calculator. This tool can find the three unknown real or imaginary roots of a polynomial equation of degree three.

It is the only cubic equation calculator available online which can provide the steps required for the roots’ calculation.

What is a cubic equation?

Like quadratic equations, it is also a polynomial equation. It has a maximum variable degree of three. This means it must have an x3.

The general cubic equation looks like this:

“ax3 + bx2 + cx + d = 0”  where a,b,c, and d are coefficients and a ≠ 0.

Some examples of cubic polynomial are:

  • 2x3 - 5x3 + 5x + 2
  • -x3 - 10
  • 24x3 - x - 12x2 + 17

Notice that each expression has x3 as the leading term.

Roots of a cubic polynomial:

The next thing after the introduction of the cubic equation is its roots. Recall that a linear equation has one root while a quadratic equation has two roots. Make a guess about “How many roots does a cubic equation have?”. Three!

There can be two cases regarding these roots.

  • All three are real.
  • One real, two complex.

A cubic polynomial always has one real root. In the second case, the complex roots are a pair of conjugates. 

Cubic polynomial formula:

The formulas used for the reduction of cubic polynomials to their roots are pretty complex. Have a look:

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

Here, b is the coefficient of x2. Complex hun? it does not end here. To find the S and T, use the formulas written below. 

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

Letters a,b,c, and d are familiar. They are coefficients of the polynomial expression. But what are Q and R? If you fear that they are to be calculated then you are right! 

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

We feel sorry for you! But no worries, our cubic equation solver is here to save the day. Just punch in the coefficients and save yourself the trouble of going through this exhausting process of long calculations.

How to solve a cubic equation?

Since the cubic formula is very long, there are several other ways in practice to solve the cubic polynomials. In this article, you will learn the formula method only. 

  1. First thing first, arrange the equation in the standard form
  2. After that write the coefficients separately.
  3. Find the Q and R values first.
  4. Then S and T.
  5. Now, use all these values to find the roots of the mentioned formulas.

Example:

Find the roots of the following cubic equation:

X3 + 2x2 + 3x + 12 = 0

Solution: 

Step 1: Arrange in the standard form.

The equation is already in the standard form. 

Step 2: Separate the coefficients.

a = 1
b = 2
c = 3
d = 12

Step 3: Find the Q and R values.

Q = (3ac - b²) / (9a²)
Q = (3(1)(3) - (2)²) / (9(1)²)
Q = 0.556
R = (9abc - 27a²d - 2b³) / (54a³)
R = (9(1)(2)(3) - 27(1)2(12) - 2(2)3) / (54(1)3)
R = -5.2962

Step 4: Find S and T.

S = ∛(R + √(Q³ + R²))
S = ∛(-5.2962) + √((0.556)³ + (-5.2962)²))
S = 0.2531
T=∛(R - √(Q³ + R²))
T=∛(-5.2962) - √((0.556)³ + (-5.2962)²))
T = 2.1973

Step 5: Lastly, use the roots formula.

X1 = First root = S + T - b/3a
X1 = First root = 0.2531 + 2.1973 - 2/3(1)
= -2.6111

X2 = Second root = -(S + T)/2 + (S - T)* i√3/2 - b/3a
X2 = Second root = -(0.2531 + 2.1973)/2 + (0.2531 - 2.1973)* i√3/2 - 2/3(1)
= -2.1218 + 2.12i

X3 = Third root = -(S + T)/2 - (S - T)* i√3/2 - b/3a
X3 = Third root = -(0.2531 + 2.1973)/2 - (0.2531 - 2.1973)* i√3/2 - 2/3(1)
= -2.1218 - 2.12i

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