Cubic Equation Solver
Solve ax³ + bx² + cx + d = 0 for all three roots, real or complex.
Roots
x = 1, x = 2, x = 3
- Equation
- x³ − 6x² + 11x − 6 = 0
- Discriminant
- 4
- Root types
- Three real roots
Show the work
Discriminant
Δ = 18abcd − 4b to the power of 3d + b to the power of 2c to the power of 2 − 4ac to the power of 3 − 27a to the power of 2d to the power of 2
18 × 1 × (−6) × 11 × (−6) − 4 × (−6) to the power of 3 × (−6) + (−6) to the power of 2 × 11 to the power of 2 − 4 × 1 × 11 to the power of 3 − 27 × 1 to the power of 2 × (−6) to the power of 2 = 4
Δ > 0: three real roots
x = 1, x = 2, x = 3
How it works
Reduce to t³ + pt + q = 0 with x = t − b/3a; three real roots use the trigonometric form, one real root uses Cardano's formula
Questions
- How does the cubic equation solver work?
- Reduce to t³ + pt + q = 0 with x = t − b/3a; three real roots use the trigonometric form, one real root uses Cardano's formula.
- What do I need to enter?
- a, b, c, d. Optional fields can be left blank.
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