Penyelesai persamaan padu khusus dengan punca sebenar dan kompleks, langkah kaedah Cardano, grafik padu dan contoh yang berfungsi.
Kalkulator Akar Kubik
Masukkan pekali polinomial anda di atas dan klik "Cari Semua Akar" untuk melihat keputusan.The roots of a cubic equation are the values of x that satisfy ax³ + bx² + cx + d = 0. By the Fundamental Theorem of Algebra, every cubic equation has exactly three roots (counting multiplicity), which may be real or complex. Graphically, real roots correspond to points where the cubic curve crosses or touches the x-axis.
Cubic roots can appear in three configurations: (1) three distinct real roots — the curve crosses the x-axis three times; (2) one real root plus two complex conjugate roots — the curve crosses the x-axis once and turns away twice; (3) repeated roots — the curve is tangent to the x-axis at one or more points. The discriminant Δ determines which case applies.
Computing cubic roots efficiently requires combining multiple strategies: the Rational Root Theorem for nice integer/fraction answers, Cardano's formula for exact algebraic expressions, and the trigonometric method for the three-real-roots case. Our calculator automatically selects the optimal strategy and delivers all three roots with full precision.
Finding where stress or deflection equations cross zero helps engineers identify critical load points in beams and frames.
Break-even analysis for cubic cost/revenue models requires finding all real roots to identify profitable operating ranges.
Cubic growth models in biology use root-finding to predict population equilibria and tipping points.
Not every cubic has three real roots. Always check the discriminant first to know what to expect.
A root can repeat (e.g., x=2 is a double root). The total count is always 3, but distinct roots may be fewer.
Intermediate calculations involve cube roots and fractions. Rounding prematurely can shift root locations significantly.
| Total Roots | Always exactly 3 (counting multiplicity) |
| Real Root Guarantee | At least 1 real root always exists |
| Complex Roots | Always appear in conjugate pairs (a±bi) |
| Methods Used | Cardano + Trigonometric + Numerical |
| Verification | Substitute each root back into the original equation |
Tidak. Setiap persamaan padu menjamin sekurang-kurangnya satu punca sebenar kerana sifat lengkung padu.
Ini bermakna lengkung berubah arah tetapi gagal melintasi paksi-x pada pusingan tertentu itu. Akar kompleks sentiasa berlaku secara berpasangan.
Punca yang berulang bermakna lengkung adalah tangen kepada paksi-x (menyentuhnya tanpa menyilang sepenuhnya).
Tepat tiga punca (mengira multiplicity). Mereka boleh menjadi tiga punca nyata yang berbeza, satu konjugat nyata dan dua konjugat kompleks, atau gabungan dengan akar berulang.
Punca sebenar ialah nilai pada garis nombor di mana lengkung melintasi atau menyentuh paksi-x. Punca kompleks melibatkan nombor khayalan dan tidak muncul sebagai pintasan-x pada graf piawai.
