Vieta’s Formulas | 华氏定理（韦达定理）及其应用

📚 Vieta’s Formulas | 华氏定理（韦达定理）及其应用

In elementary algebra, the theorem that connects the roots and coefficients of a polynomial is commonly credited to François Viète. In some Chinese reference materials it is also casually called the Hua theorem, but in the context of standard algebra examinations, the name normally refers to Vieta’s formulas.

在初等代数中，揭示多项式根与系数关系的定理通常归功于法国数学家弗朗索瓦·韦达。部分中文资料中称之为华氏定理，而在中学代数考点语境下，这个名称通常指的就是韦达定理。

1. Quadratic Equations: The Basic Pair | 一元二次方程中的基本关系

For any quadratic equation ax² + bx + c = 0 with a ≠ 0, if its two roots are x₁ and x₂, then Vieta’s formulas take the simplest form:

对于一元二次方程 ax² + bx + c = 0（a ≠ 0），设两个根为 x₁ 和 x₂，则韦达定理的最简形式为：

x₁ + x₂ = −b/a, x₁x₂ = c/a

This pair of relations holds for real roots and complex roots alike, provided that repeated roots are counted with multiplicity.

这一组关系对实数根和复数根均成立，只需按重数计数。

2. Proof by the Quadratic Formula | 用求根公式证明

Using the quadratic formula x = (−b ± √Δ)/(2a), where Δ = b² − 4ac, we can derive both relations directly.

利用求根公式 x = (−b ± √Δ)/(2a)，其中 Δ = b² − 4ac，可以一步推出两组关系。

x₁ + x₂ = (−b + √Δ + (−b − √Δ))/(2a) = −b/a

x₁x₂ = [(−b)² − (√Δ)²]/(4a²) = (b² − Δ)/(4a²) = c/a

Since Δ = b² − 4ac, the product simplifies to 4ac/(4a²) = c/a. This proof does not require Δ to be non-negative.

因为 Δ = b² − 4ac，所以乘积化简为 4ac/(4a²) = c/a。这个证明过程不要求 Δ ≥ 0。

3. Reverse Statement: Constructing the Equation | 逆定理：由两根构造方程

Conversely, if two numbers α and β satisfy α + β = S and αβ = P, then α and β are exactly the roots of the equation:

反之，若两个数 α 和 β 满足 α + β = S，αβ = P，那么 α 和 β 一定是方程的解：

x² − Sx + P = 0

This reverse form is widely used in problems that ask for a new quadratic equation with prescribed roots.

这一逆用形式在“已知两根，构造新二次方程”的试题中极为常见。

4. Sign of Roots and the Discriminant | 根的正负与判别式

Combining Vieta’s formulas with the discriminant Δ gives a complete classification of the signs of real roots.

将韦达定理与判别式 Δ 结合，可以完整判断两个实数根的符号。