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Depressed Cubic Calculator
Enter your polynomial coefficients above and click "Depress Cubic" to see results.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.
Every Cardano's method solution begins with depressing the cubic. This calculator automates that critical first step.
Depressing centers the cubic's inflection point at the y-axis, making graphical analysis and symmetry identification easier.
Students learning polynomial transformations use depression to understand how substitutions simplify equation structure.
The substitution is x = t MINUS b/(3a), not plus. Getting the sign wrong shifts the curve in the wrong direction.
The formulas for p and q assume proper normalization. If a ≠ 1, you must include a in the denominators.
p involves 3ac − b² while q involves 2b³ − 9abc + 27a²d. Mixing them up produces completely wrong depressed coefficients.
| 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 |
Because the equation has been "depressed" in complexity by stripping away the degree-two term.
Yes, it shifts them horizontally. Once you find <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">t</span>, you must add back the shift factor to find <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x</span>.
Yes. Every standard cubic equation can be shifted to eliminate its <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span> term.
The standard form is t³ + pt + q = 0, which has no squared term.
Cardano's formula only applies directly to the depressed form. By removing the squared term, the algebra becomes manageable enough to derive a closed-form solution.
