简化三次方程计算器

具有实根和复根的专用三次方程求解器、卡尔达诺方法步骤、三次图形和工作示例。

简化三次方程计算器

在上面输入您的多项式系数，然后点击“压三次”查看结果。

图形将在您求解后出现在此处。

什么是简化三次方程计算器?

简单解释：这是一个三次方程，没有x²术语，通常采用以下形式t3 + pt + q = 0.
为什么它在三次方程中很重要：消除平方项是从数学上推导一般三次方程的根的强制性第一步。它极大地简化了代数工作。

公式 / 方法

公式：代替x = t - \frac{乙}{3a}
变量解释： * p = 压裂{3ac - b²}{3a²} * q = \frac{2b³ - 9abc + 27a²d}{27a立方}* 新方程变为t3 + pt + q = 0.

➀如何使用

提供系数a、b、c、d你原来的方程。
单击“压低方程”。
查看必填项x到t替代。
通过记下您的新内容来结束p和q系数。

⭐关键特性

纯粹专注于一个重要的代数步骤。
提供准确的p和q立即输出。
快速、轻量级且数学上严格。
多项式翻译的优秀学习辅助工具。

📈 视觉图表

示例概念

原来的：x3 - 6x2 + 11x - 6 = 0工具替代品x = t + 2。产生的压抑立方：t3 - t = 0.

交互式深度分析

A depressed cubic is a cubic equation that has been simplified by removing the x² term, resulting in the cleaner form t³ + pt + q = 0. This reduction is achieved through the Tschirnhaus substitution x = t − b/(3a), which shifts the polynomial so the second-degree coefficient vanishes. The technique is fundamental to every analytical method for solving cubics.

The depression process transforms four coefficients (a, b, c, d) into just two essential parameters (p, q). Specifically: p = (3ac − b²) / (3a²) and q = (2b³ − 9abc + 27a²d) / (27a³). This simplification is what makes Cardano's formula, Vieta's substitution, and the trigonometric method feasible.

Geometrically, depressing a cubic is equivalent to horizontally translating the graph so that the inflection point sits on the y-axis. This centering eliminates asymmetry in the curve and reveals the polynomial's essential shape, making both algebraic and graphical analysis cleaner.

🎯 实际应用

🔎

Prerequisite for Cardano

Every Cardano's method solution begins with depressing the cubic. This calculator automates that critical first step.

📊

Graph Centering

Depressing centers the cubic's inflection point at the y-axis, making graphical analysis and symmetry identification easier.

🎓

Algebra Courses

Students learning polynomial transformations use depression to understand how substitutions simplify equation structure.

⚠ 常见错误及避免

1. Sign errors in the substitution

The substitution is x = t MINUS b/(3a), not plus. Getting the sign wrong shifts the curve in the wrong direction.

2. Forgetting to divide by a first

The formulas for p and q assume proper normalization. If a ≠ 1, you must include a in the denominators.

3. Confusing p and q formulas

p involves 3ac − b² while q involves 2b³ − 9abc + 27a²d. Mixing them up produces completely wrong depressed coefficients.

📋 快速参考表

Substitution	x = t − b/(3a)
Result Form	t³ + pt + q = 0
p formula	(3ac − b²) / (3a²)
q formula	(2b³ − 9abc + 27a²d) / (27a³)
Geometric Effect	Centers inflection point on y-axis

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三次方程求解器 |求解 ax³ + bx² + cx + d = 0