Cubic Equation Solver
Solve any degree-3 cubic equation ax³ + bx² + cx + d = 0 instantly. Get complete step-by-step Cardano derivations, Tschirnhaus depressed cubic reduction, exact real and complex roots, and an interactive curve graph.
Cubic Coefficients
ax³ + bx² + cx + d = 0Cardano's Method & Polynomial Reduction Breakdown
1 Depressed Cubic (t³ + pt + q = 0)
Substituting x = t - b/(3a) eliminates the quadratic term (x²):
q = 0.0000
2 Discriminant (Δ = q²/4 + p³/27)
The sign of Cardano's discriminant determines the algebraic nature of the roots:
Δ < 0 ⇒ 3 Real Roots (Viète)
3 Calculus Derivative Analysis (f')
Derivative 3ax² + 2bx + c = 0 yields turning points:
Local Min: x = 2.5774
How to Solve a Cubic Equation ax³ + bx² + cx + d = 0
A cubic equation is solved algebraically using Cardano’s method: first substitute x = t - b/(3a) to eliminate the x² term and obtain the depressed cubic t³ + pt + q = 0. Calculate Cardano’s discriminant Δ = (q/2)² + (p/3)³. If Δ > 0, there is 1 real root and 2 complex conjugates; if Δ = 0, multiple real roots; if Δ < 0 (casus irreducibilis), Viète’s trigonometric substitution yields three distinct real roots.
Cubic Equation Anatomy & Fundamental Theorem
A cubic equation is an algebraic polynomial equation of degree three with real or complex coefficients. In standard form, it is expressed as:
According to the Fundamental Theorem of Algebra, every degree-3 polynomial has exactly three roots in the complex plane, counting algebraic multiplicities. Because complex roots of real-coefficient polynomials always occur in conjugate pairs (u ± iv), an odd-degree polynomial must always cross the horizontal axis at least once, guaranteeing that at least one solution is guaranteed to be a real number.
Cardano’s Method & Tschirnhaus Depressed Reduction
Published in 1545 by Gerolamo Cardano in his landmark treatise Ars Magna (credited originally to Scipione del Ferro and Niccolò Fontana Tartaglia), the algebraic solution of the cubic begins by eliminating the quadratic term bx² using the Tschirnhaus transformation:
Substitute: x = t − b / (3a)
Yielding the Depressed Cubic: t³ + pt + q = 0
Where the parameters p and q are given by:
p = (3ac − b²) / (3a²)
q = (2b³ − 9abc + 27a²d) / (27a³)
Cardano’s Discriminant & Root Nature Classification
The nature of the three roots is determined by the sign of Cardano's polynomial discriminant Δ:
1 Real + 2 Complex Roots
The curve crosses the x-axis once and has two complex conjugate solutions.
Multiple Real Roots
All roots are real, with at least one repeated root where the curve is tangent to the axis.
3 Distinct Real Roots
The curve crosses the x-axis at three separate locations (Casus Irreducibilis).
Casus Irreducibilis & Viète’s Trigonometric Solution
When Δ < 0, evaluating Cardano’s formula requires taking the square root of a negative quantity, producing complex numbers during intermediate algebraic calculation even though all three final roots are strictly real. In 1591, François Viète discovered that this case can be resolved purely using the triple-angle cosine identity cos(3θ) = 4 cos³(θ) − 3 cos(θ):
θ = arccos((3q / (2p)) · √(−3 / p))
t_k = 2 · √(−p / 3) · cos((θ + 2kπ) / 3), for k ∈ {0, 1, 2}
x_k = t_k − b / (3a)
Calculus Analysis: Inflection Point & Critical Extrema
Applying differential calculus reveals critical features of the cubic landscape:
Point of Inflection
Setting the second derivative f''(x) = 6ax + 2b = 0 yields the point of inflection at x = -b/(3a). The curve possesses exact 180° point symmetry around this inflection coordinate.
Local Extrema (Max / Min)
Setting f'(x) = 3ax² + 2bx + c = 0 gives the critical points. If 4b² - 12ac > 0, the curve reaches one local peak and one local trough. Otherwise, the curve is strictly monotonic.
Step-by-Step Worked Algebraic Solutions
Example 1: Solve x³ - 6x² + 11x - 6 = 0
Given a = 1, b = -6, c = 11, d = -6.
Shift: x = t - (-6)/3 = t + 2
Depressed parameters: p = (33 - 36)/3 = -1, q = 0
Depressed equation: t³ - t = 0 ⇒ t(t - 1)(t + 1) = 0
Solutions for t: t = -1, 0, 1
Substitute back: x = t + 2 ⇒ x₁ = 1, x₂ = 2, x₃ = 3
Example 2: Complex root cubic x³ - 3x² + 4x - 12 = 0
Given a = 1, b = -3, c = 4, d = -12.
Factor by grouping: x²(x - 3) + 4(x - 3) = (x² + 4)(x - 3) = 0
Roots: x₁ = 3 (real), x₂ = +2i, x₃ = -2i (complex conjugates)
Common Pitfalls & Calculation Errors
If a graphing calculator only shows one x-intercept, students often believe the equation only has one solution. Every cubic has 3 roots; if only one real intersection is visible, the other two roots are complex numbers.
If the leading coefficient a = 0, the degree drops to 2, reducing the equation to a quadratic form bx² + cx + d = 0, which must be solved using the quadratic formula.
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