Pemecah persamaan kubik khusus dengan akar nyata dan kompleks, langkah-langkah metode Cardano, grafik kubik, dan contoh kerja.
Kalkulator Kubik Tertekan
Masukkan koefisien polinomial Anda di atas dan klik "Tekan Kubik" untuk melihat hasilnya.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 |
Karena persamaan tersebut telah "ditekan" kompleksitasnya dengan menghilangkan suku derajat dua.
Ya, itu menggesernya secara horizontal. Setelah Anda menemukannya<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">T</span>, Anda harus menambahkan kembali faktor pergeseran untuk menemukannya<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">X</span>.
Ya. Setiap persamaan kubik standar dapat digeser untuk menghilangkannya<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span>ketentuan.
Bentuk standarnya adalah t³ + pt + q = 0, yang tidak memiliki suku kuadrat.
Rumus Cardano hanya berlaku langsung pada bentuk depresi. Dengan menghilangkan suku kuadrat, aljabar menjadi cukup mudah dikelola untuk mendapatkan solusi bentuk tertutup.
