Dedikerad kubisk ekvationslösare med verkliga och komplexa rötter, Cardano-metodsteg, kubikgrafer och utarbetade exempel.
Deprimerad kubikräknare
Ange dina polynomkoefficienter ovan och klicka på "Tryck ner Cubic" för att se resultat.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 |
Eftersom ekvationen har blivit "deprimerad" i komplexitet genom att ta bort grad-två-termen.
Ja, det flyttar dem horisontellt. När du hittar<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">t</span>, måste du lägga tillbaka skiftfaktorn för att hitta<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x</span>.
Ja. Varje vanlig kubikekvation kan skiftas för att eliminera dess<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span>kalla.
Standardformen är t³ + pt + q = 0, som inte har någon kvadratterm.
Cardanos formel gäller bara direkt för den deprimerade formen. Genom att ta bort den kvadratiska termen blir algebra hanterbar nog att härleda en sluten formlösning.
