Depressed Cubic Calculator
Depressed Cubic Calculator. Find real and complex roots, discriminant, and cubic graphs in seconds with our precision solver.
Depressed Cubic Calculator
Enter your polynomial coefficients above and click "Depress Cubic" to see results.What is Depressed Cubic Calculator?
- Simple explanation: It is a cubic equation that has no x² term, usually taking the form t³ + pt + q = 0.
- Why it matters in cubic equations: Eliminating the squared term is the mandatory first step in mathematically deriving the roots of a generic cubic equation. It vastly simplifies algebraic work.
Formula / Method
- Formula: Substitute x = t - \frac{b}{3a}
- Variables Explained: * p = \frac{3ac - b²}{3a²} * q = \frac{2b³ - 9abc + 27a²d}{27a³} * The new equation becomes t³ + pt + q = 0.
How To Use
- Provide the coefficients a, b, c, d of your original equation.
- Click "Depress Equation."
- View the mandatory x to t substitution.
- Finalize by noting your new p and q coefficients.
Key Features
- Focuses purely on one vital algebraic step.
- Provides the exact p and q outputs instantly.
- Fast, lightweight, and mathematically strict.
- Excellent learning aid for polynomial translation.
Example Concept
Original: x³ - 6x² + 11x - 6 = 0 The tool substitutes x = t + 2. Resulting Depressed Cubic: t³ - t = 0.
Interactive Deep Dive
A depressed cubic is a cubic equation that has been simplified by removing the x² term, resulting in the cleaner form t³ + pt + q = 0. This reduction is achieved through the Tschirnhaus substitution x = t − b/(3a), which shifts the polynomial so the second-degree coefficient vanishes. The technique is fundamental to every analytical method for solving cubics.
The depression process transforms four coefficients (a, b, c, d) into just two essential parameters (p, q). Specifically: p = (3ac − b²) / (3a²) and q = (2b³ − 9abc + 27a²d) / (27a³). This simplification is what makes Cardano's formula, Vieta's substitution, and the trigonometric method feasible.
Geometrically, depressing a cubic is equivalent to horizontally translating the graph so that the inflection point sits on the y-axis. This centering eliminates asymmetry in the curve and reveals the polynomial's essential shape, making both algebraic and graphical analysis cleaner.
Visual Diagram
Depressed Cubic Transformation — Removing the x² term simplifies everything
Real-World Applications
Prerequisite for Cardano
Every Cardano's method solution begins with depressing the cubic. This calculator automates that critical first step.
Graph Centering
Depressing centers the cubic's inflection point at the y-axis, making graphical analysis and symmetry identification easier.
Algebra Courses
Students learning polynomial transformations use depression to understand how substitutions simplify equation structure.
Common Mistakes to Avoid
1. Sign errors in the substitution
The substitution is x = t MINUS b/(3a), not plus. Getting the sign wrong shifts the curve in the wrong direction.
2. Forgetting to divide by a first
The formulas for p and q assume proper normalization. If a ≠ 1, you must include a in the denominators.
3. Confusing p and q formulas
p involves 3ac − b² while q involves 2b³ − 9abc + 27a²d. Mixing them up produces completely wrong depressed coefficients.
Quick Reference Table
| Substitution | x = t − b/(3a) |
| Result Form | t³ + pt + q = 0 |
| p formula | (3ac − b²) / (3a²) |
| q formula | (2b³ − 9abc + 27a²d) / (27a³) |
| Geometric Effect | Centers inflection point on y-axis |
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