Cubic Equations
Cubic equations ax³+bx²+cx+d=0 have up to three roots. Cardano's formula (16th century) solves them via the depressed cubic y³+py+q=0. Discriminant Δ determines whether roots are real or complex.
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Depressed cubic removes the x² term for easier solution. When Δ<0, all three roots are real (casus irreducibilis). Sum of roots = −b/a, product = −d/a (Vieta).
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Why: Cubics arise in optimization, geometry (angle trisection), and physics. The solution formula marked a breakthrough in Renaissance algebra. Vieta's relations link roots to coefficients.
How: Substitute x=y−b/(3a) to depress the cubic. Cardano: solve y³+py+q=0 using u³+v³=−q and uv=−p/3. Roots come from cube roots of u³ and v³. Discriminant Δ=q²/4+p³/27.
Run the calculator when you are ready.
Cubic Equation — Cardano's Method
Solve ax³+bx²+cx+d=0. All roots, discriminant, Vieta's relations.
Root Values
Real vs Complex Roots
Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
Cardano published the formula in 1545; Tartaglia discovered it.
— History
Vieta: r₁+r₂+r₃=−b/a, r₁r₂r₃=−d/a.
— Relations
Key Takeaways
- • Every cubic equation ax³ + bx² + cx + d = 0 has exactly three roots (counting multiplicity), by the Fundamental Theorem of Algebra
- • The discriminant Δ = q²/4 + p³/27 determines the nature: Δ < 0 → three real roots; Δ > 0 → one real + two complex; Δ = 0 → repeated roots
- • Cardano's formula (1545) solves the depressed cubic y³ + py + q = 0 via substitution and cube roots
- • Vieta's relations: sum of roots = -b/a, product = -d/a — useful for verification
- • Complex roots always appear in conjugate pairs when coefficients are real
- • The depressed cubic eliminates the x² term via x = y - b/(3a), simplifying the algebra
Did You Know?
How It Works
Cardano's method transforms ax³+bx²+cx+d=0 into a depressed cubic y³+py+q=0 by substituting x = y - b/(3a). The discriminant Δ = q²/4 + p³/27 then determines which formula to apply: for Δ<0 we use trigonometric functions (cosine of one-third angle); for Δ>0 we use cube roots of (-q/2 ± √Δ); for Δ=0 we have simplified formulas for repeated roots.
Step-by-Step Algorithm
- Divide by a to get x³ + (b/a)x² + (c/a)x + (d/a) = 0
- Substitute x = y − b/(3a) to eliminate the x² term
- Obtain depressed cubic: y³ + py + q = 0 where p = c − b²/3, q = 2b³/27 − bc/3 + d
- Compute Δ = q²/4 + p³/27
- If Δ < 0: use y = 2√(−p³/27) cos((1/3)arccos(−q/(2r))) with r = √(−p³/27)
- If Δ > 0: use y = ∛(−q/2 + √Δ) + ∛(−q/2 − √Δ)
- If Δ = 0: use simplified formulas for double/triple roots
- Convert back: x = y − b/(3a) for each root
Expert Tips
Check for Rational Roots First
Use the Rational Root Theorem: try ±(factors of d)/(factors of a). If you find one root r, factor out (x-r) and solve the resulting quadratic.
Discriminant Interpretation
Δ < 0 means three distinct real roots — the cubic crosses the x-axis three times. Δ > 0 means one real and two complex — only one crossing.
Numerical Stability
For certain coefficients, direct application of Cardano's formula can suffer from catastrophic cancellation. This calculator uses stable implementations.
Connection to Quadratics
If a=0, the equation becomes quadratic. Always verify a≠0 before applying cubic methods.
Reference Table — Discriminant & Roots
| Δ | Nature of Roots | Graph Behavior |
|---|---|---|
| Δ < 0 | Three distinct real roots | Crosses x-axis 3 times |
| Δ > 0 | One real, two complex conjugate | Crosses x-axis once |
| Δ = 0, p≠0 | One single + one double root | Tangent at double root |
| Δ = 0, p=0 | Triple root | Inflection point on x-axis |
FAQ
What is Cardano's formula?
Cardano's formula (1545) is an algebraic solution for cubic equations. It reduces the general cubic to a depressed form and uses cube roots. The formula works for all cubics but can produce complex intermediate expressions even when roots are real.
Why does Δ < 0 give three real roots?
When Δ < 0, the expression under the cube root becomes negative. Using complex numbers and De Moivre's theorem, we take the cosine of one-third of an angle — which always yields three distinct real values.
What are Vieta's relations?
For ax³+bx²+cx+d=0 with roots r₁,r₂,r₃: r₁+r₂+r₃=-b/a, r₁r₂+r₂r₃+r₁r₃=c/a, r₁r₂r₃=-d/a. These help verify solutions and connect coefficients to symmetric functions of roots.
When do I get complex roots?
When the discriminant Δ > 0, one root is real and the other two are complex conjugates (a±bi). This happens when the cubic has only one x-intercept.
How is the depressed cubic used?
Substituting x = y - b/(3a) eliminates the x² term, giving y³+py+q=0. This simpler form makes the algebra tractable. The transformation preserves the root structure.
Can I use this for quartics?
Quartic equations have a more complex solution (Ferrari's method). This calculator is for cubics only. See our polynomial calculators for higher degrees.
Quick Reference
Applications in Science & Engineering
Physics
Van der Waals equation of state, eigenvalue problems in quantum mechanics, cubic potential wells, and oscillatory systems with cubic nonlinearities.
Engineering
Beam deflection equations, control system characteristic polynomials, cubic spline interpolation, and fluid dynamics models.
Disclaimer: This calculator uses Cardano's method for educational purposes. Results are numerically stable for typical coefficients. For research or high-precision work, verify with symbolic algebra systems (Mathematica, Maple, SymPy).
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