Cubic Equation Calculator

The Cubic Equation Calculator helps you solve cubic equations of the form:

$$ax^3 + bx^2 + cx + d = 0$$

where:

• $a$, $b$, $c$, and $d$ are real coefficients of the equation.
• $x$ represents the variable, and the calculator provides up to three real or complex solutions for $x$.

How to Use:

1. Enter Coefficients:

   • In the input fields, enter values for the coefficients $a$, $b$, $c$, and $d$. The coefficient $a$ must be non-zero because a cubic equation requires the $x^3$ term.
   • Use the +/− button to easily toggle between positive and negative values for each coefficient.

2. Select Significant Figures:

   • If you want a specific precision for your answers, adjust the significant figures setting (optional).

3. Click “Calculate”:

   • Once you’ve entered all values, click the “Calculate” button to compute the solutions.

4. View Solutions:

   • The calculator will display the real or complex solutions for $x$. If the equation has complex solutions, they will be shown in the form $a + bi$.

5. Clear Fields:

   • Use the “Clear” button to reset all input fields and start a new calculation.

Example

Let’s solve the cubic equation:

$$2x^3 – 4x^2 + 3x – 6 = 0$$

1. Enter Values:

   • $a = 2$
     
   • $b = -4$
     
   • $c = 3$
     
   • $d = -6$
     

2. Click “Calculate”: The calculator processes the inputs and solves the cubic equation.

3. View Results:

   • You’ll see the following results:
     • $x_1 = 2.540$
       
     • $x_2 = -0.740 + 1.561i$
       
     • $x_3 = -0.740 – 1.561i$
       

Formulas

The calculator uses the following approach to solve the cubic equation:

1. Depressed Cubic Transformation: The cubic equation is simplified using the formula:

$$p = \frac{3ac – b^2}{3a^2}, \quad q = \frac{2b^3 – 9abc + 27a^2d}{27a^3}$$

2. Discriminant: The discriminant is calculated to determine the nature of the roots:

$$\Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3$$

• If $\Delta > 0$, there is one real root and two complex conjugate roots.
• If $\Delta = 0$, there are multiple real roots.
• If $\Delta < 0$, all three roots are real.

3. Solving for Roots:

   • For real roots, the solutions are determined by:
   $$x_1 = 2\sqrt[3]{r}\cos\left(\frac{\theta}{3}\right) – \frac{b}{3a}$$

   Where $r = -\frac{q}{2}$ and $\theta$ is the angle calculated based on the discriminant.

4. Complex Roots: If the discriminant is negative, the complex roots are computed using trigonometric or hyperbolic functions, resulting in expressions like $x_2 = a + bi$.

Quick Tips

• Make sure $a$ is non-zero; otherwise, the equation is not cubic.
• Complex solutions are shown in the form $x = a + bi$, where $i$ is the imaginary unit.
• The calculator can handle both positive and negative coefficients, and you can easily toggle the sign using the +/− button.