Penyelesai persamaan padu khusus dengan punca sebenar dan kompleks, langkah kaedah Cardano, grafik padu dan contoh yang berfungsi.
Kalkulator Kaedah Cardano
Masukkan pekali polinomial anda di atas dan klik "Gunakan Kaedah Cardano" untuk melihat keputusan.Cardano's method, published by Gerolamo Cardano in 1545, is the first known general algebraic solution for cubic equations. The method works by transforming the general cubic ax³ + bx² + cx + d = 0 into a depressed cubic (one without the x² term) using the substitution x = t − b/(3a). The resulting equation t³ + pt + q = 0 is simpler to solve algebraically.
The solution relies on a clever decomposition: set t = u + v, which leads to the system u³ + v³ = −q and uv = −p/3. Solving this system produces the discriminant Δ = q²/4 + p³/27. When Δ > 0, the cube roots are straightforward. When Δ < 0, the famous casus irreducibilis occurs — all three roots are real, yet the formula requires passage through complex numbers.
Despite its age, Cardano's method remains the foundation of cubic equation theory and is still taught in advanced algebra courses worldwide. Understanding it builds essential bridges to group theory, Galois theory, and the deeper question of why no analogous formula exists for polynomials of degree five or higher.
Cardano's method is a cornerstone of university algebra curricula, teaching students how general polynomial solutions are derived.
Characteristic equations of third-order systems are solved analytically using Cardano's approach for exact pole placement.
Exact cubic solutions are needed in optics (Snell's law extensions), fluid dynamics, and orbital mechanics calculations.
You must eliminate the x² term first. Applying Cardano's formula directly to the general form yields incorrect results.
When Δ < 0, the formula involves complex cube roots even though all roots are real. Use trigonometric substitution instead.
The depressed coefficients involve fractions with 3a, 27a³, etc. Double-check these intermediate values carefully.
| Substitution | x = t − b/(3a) |
| Depressed Form | t³ + pt + q = 0 |
| Discriminant | Δ = q²/4 + p³/27 |
| Published | 1545 by Gerolamo Cardano |
| Limitation | Casus irreducibilis when Δ < 0 |
Ia mencecah halangan yang dipanggil "casus irreducibilis" apabila terdapat tiga punca sebenar (Δ > 0). Semasa fasa ini, ia memerlukan nombor kompleks untuk mencari jawapan sebenar.
Tidak, kalkulator melakukan secara automatik<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">t - b/3a</span>pengganti untuk anda.
Tidak, kaedah trigonometri sering diutamakan apabila tiga punca sebenar wujud.
Ia telah diterbitkan oleh Gerolamo Cardano pada tahun 1545 dalam bukunya Ars Magna, walaupun teknik asasnya sebahagiannya ditemui oleh Scipione del Ferro dan Niccolò Tartaglia.
Ya, ia direka khusus supaya anda boleh mengikuti dan mempelajari kaedah dan bukannya menyalin jawapan sahaja.
