Step-by-step Instructions

1. Enter the coefficients of your cubic equation in the provided input fields.
2. Ensure you provide coefficients for all terms, even if they're zero.
3. Click on the "Calculate" button.
4. The solutions, along with a breakdown based on the discriminant, will appear below.
5. If your equation has complex roots, they will be presented in the form of a + bi.

Example Card

Solving Cubic Equations: A Breakdown

1. Case when \( \Delta > 0 \)

Description: One real root and two non-real complex conjugate roots.

Example: \( x^3 - 6x^2 + 11x - 6 = 0 \)

• Normalize the equation (make sure the coefficient of \( x^3 \) is 1).
• Calculate the discriminant: \[ \Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2 \]
• Given \( \Delta > 0 \), there will be one real root and two non-real complex conjugate roots.
• Find the real root using appropriate methods.
• The complex roots can be found using polynomial division once the real root is identified.

2. Case when \( \Delta = 0 \)

Description: All roots are real, and at least two are equal.

Example: \( x^3 - 3x^2 + 3x - 1 = 0 \)

• Normalize the equation.
• Calculate the discriminant.
• Given \( \Delta = 0 \), there will be three real roots, and at least two of them will be the same.
• The repeated root can be found using a formula or by inspection.
• Polynomial division will help find the third root.

3. Case when \( \Delta < 0 \)

Description: Three distinct real roots.

Example: \( x^3 - 15x - 4 = 0 \)

• Normalize the equation.
• Calculate the discriminant.
• Given \( \Delta < 0 \), all three roots will be distinct and real.
• The roots can be found using Cardano's method.

Case when \( \Delta > 0 \):

Example Equation: \( x^3 - 3x^2 + x + 1 = 0 \)

Solutions:

• One real root: \( x_1 = 2 \)
• Two non-real complex conjugate roots: \( x_2 = 0.5 + i\sqrt{2} \) and \( x_3 = 0.5 - i\sqrt{2} \)

Case when \( \Delta = 0 \):

Example Equation: \( x^3 - 3x^2 + 3x - 1 = 0 \)

Solutions:

• Three real roots, with two of them being the same: \( x_1 = x_2 = 1 \) and \( x_3 = 1 \)

Case when \( \Delta < 0 \):

Example Equation: \( x^3 - 6x^2 + 11x - 6 = 0 \)

Solutions:

• Three distinct real roots: \( x_1 = 1 \), \( x_2 = 2 \), and \( x_3 = 3 \)