Cubic Equation Calculator

Solve cubic equations in the form ax³ + bx² + cx + d = 0 using the cubic formula. This calculator shows you all the steps and finds all real solutions.

Equation:

1x³ + 0x² + 0x + 0 = 0

Tips:

• Enter coefficients for the equation ax³ + bx² + cx + d = 0

• Coefficient 'a' cannot be zero (not a cubic equation if a=0)

• The calculator will show step-by-step solution process

• Solutions include both real and complex roots

How It Works

The cubic formula is used to find the roots of a third-degree polynomial equation. The process involves these main steps:

1. Normalize equation (divide by a if a ≠ 1)

2. Calculate intermediate values (p, q, r)

3. Apply cubic formula to find roots

4. Determine number of real solutions

Example Solutions

Simple Example

x³ - 6x² + 11x - 6 = 0

  1. Already in standard form (a = 1)
  2. Calculate discriminant values
  3. Apply cubic formula
  4. Solutions: x = 1, x = 2, x = 3

Real-World Applications

Physics: Volume Expansion

A cube's volume increases by temperature following a cubic equation:

V = V₀(1 + αT)³

Find temperature for specific volume:

(1 + αT)³ = V/V₀

Engineering: Beam Deflection

The deflection of a loaded beam can follow a cubic equation:

y = ax³ + bx² + cx + d

Find points of maximum deflection:

dy/dx = 3ax² + 2bx + c = 0

Types of Solutions

  • One real root and two complex conjugate roots
  • Three real roots (may include repeated roots)
  • Three distinct real roots

Tips for Success

  • Always check if the equation can be factored first
  • Look for rational roots using the rational root theorem
  • Verify solutions by substituting back into the original equation
  • Remember that a cubic equation always has at least one real root
  • Use graphing to estimate the number of real roots