Penyelesai persamaan padu khusus dengan punca sebenar dan kompleks, langkah kaedah Cardano, grafik padu dan contoh yang berfungsi.
Kalkulator Akar Kompleks
Masukkan pekali polinomial anda di atas dan klik "Cari Akar Kompleks" untuk melihat keputusan.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) |
Selagi pekali asal polinomial ialah nombor nyata, akar kompleks mesti datang sebagai "konjugat" (satu tambah, satu tolak) supaya bahagian kompleksnya terbatal apabila dirumuskan semula.
Tidak. Oleh kerana lengkung padu mempunyai satu hujung naik selama-lamanya dan hujung satu lagi turun selama-lamanya, ia mesti melintasi paksi nyata mendatar sekurang-kurangnya sekali.
Bahagian khayalan mewakili sejauh mana punca dari garis nombor sebenar dalam satah kompleks. Ia tidak mempunyai persimpangan paksi-x fizikal tetapi penting untuk algebra berfungsi.
Konjugat kompleks mempunyai bahagian nyata yang sama tetapi bahagian khayalan bertentangan. Jika satu punca ialah a + bi, yang satu lagi ialah a - bi.
Akar kompleks tidak menghasilkan persilangan paksi-x yang kelihatan pada graf. Mereka mempengaruhi bentuk lengkung dalam satah sebenar tetapi wujud di luar skrin dalam satah kompleks.
