Pemecah persamaan kubik khusus dengan akar nyata dan kompleks, langkah-langkah metode Cardano, grafik kubik, dan contoh kerja.
Kalkulator Akar Kompleks
Masukkan koefisien polinomial Anda di atas dan klik "Temukan Akar Kompleks" untuk melihat hasilnya.Complex roots are solutions to polynomial equations that involve the imaginary unit i = √(−1). They take the form a + bi, where a is the real part and b is the imaginary part. For cubic equations with real coefficients, complex roots always appear in conjugate pairs: if a+bi is a root, then a−bi must also be a root.
A cubic equation has complex roots when its discriminant Δ < 0. In this case, there is exactly one real root and two complex conjugate roots. On the graph, the real root appears as an x-axis crossing, while the complex roots have no visible graphical representation on the real plane — they exist in the complex plane (Argand diagram).
Complex roots are not merely mathematical curiosities. In electrical engineering, they represent oscillatory behavior in circuits. In control theory, complex poles determine system oscillation frequency and damping. In quantum mechanics, complex numbers are fundamental to wave function descriptions. This calculator extracts and displays complex roots with both rectangular (a+bi) and polar forms.
Complex roots of impedance equations determine resonant frequencies and damping behavior in AC circuits.
Complex poles of transfer functions control oscillation frequency and stability margins of feedback systems.
Filter design relies on complex root placement to achieve desired frequency response characteristics.
Complex roots of polynomials with real coefficients ALWAYS come in conjugate pairs. If you find a+bi, the other must be a−bi.
Complex roots do NOT appear on standard real-number graphs. They exist in the complex plane only.
When Δ < 0, there is still exactly one real root. Don't overlook it while focusing on the complex pair.
| Form | a + bi where i = √(−1) |
| Conjugate Rule | If a+bi is a root, so is a−bi |
| When They Appear | Discriminant Δ < 0 |
| Count | 1 real root + 2 complex conjugates |
| Polar Form | r·(cosθ + i·sinθ) |
Selama koefisien asli polinomial adalah bilangan real, akar-akar kompleks harus berbentuk "konjugat" (satu tambah, satu minus) sehingga bagian kompleksnya akan hilang ketika dirumuskan kembali.
Tidak. Karena kurva kubik memiliki satu ujung yang naik selamanya dan ujung lainnya turun selamanya, kurva tersebut harus melintasi sumbu real horizontal setidaknya satu kali.
Bagian imajiner menunjukkan jarak akar dari garis bilangan real pada bidang kompleks. Ia tidak memiliki perpotongan sumbu x secara fisik tetapi penting agar aljabar dapat berfungsi.
Konjugat kompleks mempunyai bagian nyata yang sama tetapi bagian imajinernya berlawanan. Jika salah satu akarnya adalah a + bi, akar lainnya adalah a - bi.
Akar kompleks tidak menghasilkan perpotongan sumbu x yang terlihat pada grafik. Mereka mempengaruhi bentuk kurva pada bidang nyata tetapi ada di luar layar pada bidang kompleks.
