Penyelesai persamaan padu khusus dengan punca sebenar dan kompleks, langkah kaedah Cardano, grafik padu dan contoh yang berfungsi.
Kalkulator Kubik Tertekan
Masukkan pekali polinomial anda di atas dan klik "Tekan Kubik" untuk melihat keputusan.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 |
Kerana persamaan telah "tertekan" dalam kerumitan dengan menanggalkan sebutan darjah dua.
Ya, ia mengalihkannya secara mendatar. Sebaik sahaja anda menjumpai<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">t</span>, anda mesti menambah kembali faktor anjakan untuk mencari<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x</span>.
ya. Setiap persamaan padu piawai boleh dianjak untuk menghapuskannya<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span>istilah.
Bentuk piawai ialah t³ + pt + q = 0, yang tidak mempunyai sebutan kuasa dua.
Formula Cardano hanya terpakai terus kepada bentuk yang ditekan. Dengan mengalih keluar sebutan kuasa dua, algebra menjadi cukup terurus untuk memperoleh penyelesaian bentuk tertutup.
