Ax3 Bx2 Cx D 0 Calculator

What is a cubic equation?

A cubic equation is a polynomial equation of degree three. The general form is:

Where a, b, c, and d are real numbers, and a ≠ 0. Cubic equations can have one real root and two complex conjugate roots, or three real roots (which may be equal).

Cubic equations appear in many real-world problems, including physics, engineering, economics, and biology. They are more complex than quadratic equations but can be solved using algebraic methods.

How to solve ax³ + bx² + cx + d = 0

The solution to the cubic equation can be found using Cardano's formula. Here's a step-by-step explanation:

Step 1: Depressed cubic

First, we transform the equation into a depressed cubic by substituting x = y - b/(3a). This eliminates the x² term.

Step 2: Discriminant

Calculate the discriminant Δ to determine the nature of the roots:

Step 3: Solve based on discriminant

Depending on the value of Δ, the roots are calculated differently:

• If Δ > 0: One real root and two complex conjugate roots
• If Δ = 0: Three real roots, at least two are equal
• If Δ < 0: Three distinct real roots

Example calculation

Let's solve x³ - 6x² + 11x - 6 = 0:

1. Depressed cubic: y³ + py + q = 0 where p = -4 and q = -10
2. Discriminant Δ = -108 (Δ < 0)
3. Roots: x = 1, x = 2, x = 3

Real-world applications

Cubic equations appear in various fields:

• Physics: Modeling projectile motion with air resistance
• Engineering: Designing beams and columns
• Economics: Cost-benefit analysis
• Biology: Population growth models

Limitations of this calculator

It assumes the coefficients are real numbers. Complex coefficients would require a different approach.

The calculator may not handle all edge cases perfectly due to floating-point arithmetic limitations.