Специализиран инструмент за решаване на кубични уравнения с реални и комплексни корени, стъпки на метода Cardano, кубични графики и работещи примери.
Калкулатор за кубичен дискриминант
Въведете вашите полиномни коефициенти по-горе и щракнете върху „Изчислете дискриминанта“, за да видите резултатите.The discriminant of a cubic equation is a single numerical value, denoted Δ, that instantly reveals the nature of all three roots without solving the equation. For the general cubic ax³ + bx² + cx + d = 0, the discriminant formula is Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d². This expression encodes the geometric relationship between the curve and the x-axis.
When Δ > 0, the cubic has three distinct real roots — the curve crosses the x-axis at three separate points. When Δ = 0, at least two roots coincide, meaning the curve is tangent to the x-axis at one or more points. When Δ < 0, there is exactly one real root and a pair of complex conjugate roots — the curve crosses the x-axis only once.
The discriminant serves as a classification gateway in algebra and engineering. Before investing effort into solving, you can determine whether to expect real or complex solutions. This is especially valuable in structural analysis, control systems, and optimization problems where the nature of roots determines system stability.
Determine whether a control system has oscillatory (complex) or overdamped (real) behavior based on its characteristic cubic equation.
Predict phase transitions modeled by cubic free-energy equations — the discriminant reveals whether single or multiple phases coexist.
Assess whether profit models have multiple break-even points or a single crossing, guiding pricing strategy decisions.
The discriminant formula is long. Omitting the last term is common and leads to incorrect root-type classification.
The quadratic discriminant b²−4ac is simpler. Don't accidentally use it for cubics — they require the full 5-term expression.
For quadratics, Δ>0 means 2 real roots. For cubics, Δ>0 means 3 real roots. Keep the conventions straight.
| Formula | Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d² |
| Δ > 0 | Three distinct real roots |
| Δ = 0 | At least two equal real roots |
| Δ < 0 | One real root, two complex conjugate roots |
| Input | Coefficients a, b, c, d |
| Output | Discriminant value + root classification |
Не, само ви казва какъв *тип* корени съществува. Имате нужда от специализиран решаващ инструмент, за да намерите точните стойности.
Просто въведете 0 за<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">b</span>коефициент. Калкулаторът лесно обработва липсващите термини.
Да, дискриминант нула означава, че кривата просто докосва оста x, което води до повтарящ се (множествен) корен.
Положителен дискриминант (Δ > 0) означава, че кубичното уравнение има три различни реални корена.
Отрицателен дискриминант (Δ < 0) означава, че кубичното уравнение има един реален корен и два комплексно спрегнати корена.
