# 一元二次方程根的相关函数

> 爱德思国际GCSE进阶纯数学 · 4PM1 S2
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-further-pure-math-s2-functions-of-the-roots-of/

本指南涵盖爱德思IGCSE进阶纯数学4PM1中二次方程根相关的核心技能：使用根的和与积恒等式、计算对称函数、构造具有变换根的新二次方程。

**先修:** [二次方程求解与判别式](https://www.owlsprep.com/zh/study/edexcel-igcse-further-pure-math-s2-quadratic-discriminant-solving/)

## 学习目标

- 识记并应用方程$ax^2 + bx + c = 0$的根的和（$\alpha + \beta = -b/a$）与积（$\alpha\beta = c/a$）恒等式
- 计算常见的根的对称函数，包括$\alpha^2 + \beta^2$、$1/\alpha + 1/\beta$、$\alpha^3 + \beta^3$以及$(\alpha - \beta)^2$
- 使用标准形式$x^2 - (和)x + 积 = 0$构造具有变换后根的新二次方程

## 二次方程根的和与积恒等式

对于任意形如$ax^2 + bx + c = 0$、根为$\alpha$和$\beta$的二次方程，有两个核心恒等式将二次方程的系数与其根关联起来。这些公式不会在考试公式表中给出，你必须熟记。

**二次方程根的和与积** — 若$ax^2 + bx + c = 0$的根为$\alpha, \beta$：<br> 根的和：$\alpha + \beta = -\frac{b}{a}$ <br> 根的积：$\alpha\beta = \frac{c}{a}$

*例:* For $2x^2 + 5x - 3 = 0$, $\alpha + \beta = -5/2$, $\alpha\beta = -3/2$

**例题:** 求方程$3x^2 -7x + 2 = 0$的根的和与积。

1. 步骤1：识别系数：$a=3$，$b=-7$，$c=2$
2. 步骤2：应用根的和公式：
3. $$\alpha + \beta = -\frac{b}{a} = -\left(\frac{-7}{3}\right) = \frac{7}{3}$$
4. 步骤3：应用根的积公式：
5. $$\alpha\beta = \frac{c}{a} = \frac{2}{3}$$

> **warning**
>
> 计算根的和时务必注意$b$的符号：忘记$\alpha + \beta = -b/a$中的前置负号是本知识点最常见的错误。

*计算器:* allowed

## 计算根的对称函数

根的对称函数指交换$\alpha$和$\beta$后取值不变的表达式。你无需算出单个根的具体值即可计算这类表达式：所有对称函数都可以仅通过根的和与积恒等式改写。

- $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$
- $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}$
- $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$
- $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$

**例题:** 已知$x^2 -4x +1 = 0$的根为$\alpha, \beta$，求$\alpha^3 + \beta^3$的值。

1. 步骤1：计算原方程根的和与积：
2. $$\alpha + \beta = 4, \quad \alpha\beta = 1$$
3. 步骤2：代入立方和恒等式：
4. $$\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$$
5. $$= 4^3 - 3(1)(4) = 64 - 12 = 52$$

> **考试提示**
>
> 这类题目不要浪费时间求解$\alpha$和$\beta$的具体值：使用对称恒等式更快，出错概率也低得多。

*计算器:* allowed

## 构造具有变换根的新二次方程

一旦算出一组变换后根的和与积，你就可以使用标准形式写出对应这组根的完整二次方程。

**由根构造二次方程** — 若一个二次方程的根为$p$和$q$，其标准形式为：<br> $x^2 - (p + q)x + pq = 0$ <br> 如果要求系数为整数，可以将整式乘以常数消去分母。

*例:* A quadratic with roots 3 and 5 has equation $x^2 - 8x +15 = 0$

**例题:** 已知二次方程$2x^2 +3x -2 =0$的根为$\alpha, \beta$，构造一个系数为整数、根为$2\alpha +1$和$2\beta +1$的二次方程。

1. 步骤1：计算原方程根的和与积：
2. $$\alpha + \beta = -\frac{3}{2}, \quad \alpha\beta = -1$$
3. 步骤2：计算新根的和$S$：
4. $$S = (2\alpha +1) + (2\beta +1) = 2(\alpha + \beta) + 2 = 2\left(-\frac{3}{2}\right) + 2 = -1$$
5. 步骤3：计算新根的积$P$：
6. $$P = (2\alpha +1)(2\beta +1) = 4\alpha\beta + 2(\alpha + \beta) +1 = 4(-1) + 2\left(-\frac{3}{2}\right) +1 = -6$$
7. 步骤4：代入二次方程标准形式：
8. $$x^2 - Sx + P = x^2 +x -6 = 0$$

> **tip**
>
> 务必注意题目是否要求系数为整数：将所有项乘以所有分数分母的最小公倍数，即可得到符合要求的整数系数。

*计算器:* allowed

## 常见错误

- **错误做法:** Using $\alpha + \beta = b/a$ instead of $-b/a$
  - 原因: Forgetting the leading negative sign in the sum of roots identity invalidates all subsequent calculations
  - 正确做法: Explicitly memorize the identity with the negative sign, and double-check the sign of $b$ when substituting values
- **错误做法:** Calculating individual $\alpha$ and $\beta$ values to evaluate symmetric functions
  - 原因: This is time-consuming and increases risk of arithmetic errors, especially with irrational roots
  - 正确做法: Always rewrite symmetric functions using only the sum and product of roots identities
- **错误做法:** Writing new quadratics as $x^2 + (sum)x + product = 0$
  - 原因: Mixing up the sign of the $x$ term when forming a quadratic from roots leads to an incorrect final equation
  - 正确做法: Recall the standard form: subtract the sum of roots, add the product of roots
- **错误做法:** Leaving fractions in the final quadratic when integer coefficients are required
  - 原因: Exam questions often explicitly require integer coefficients, so un-cleared fractions lose marks even if sum/product are correct
  - 正确做法: Multiply all terms by the lowest common denominator of any fractions to get integer coefficients
- **错误做法:** Expanding $(\alpha - \beta)^2$ as $\alpha^2 - \beta^2$
  - 原因: Confusing the square of a difference with the difference of squares leads to an incorrect symmetric identity
  - 正确做法: Use the pre-derived identity $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$ to avoid expansion errors

## 速查表

| Identity Name | Formula |
| --- | --- |
| Sum of roots | $\alpha + \beta = -\frac{b}{a}$ |
| Product of roots | $\alpha\beta = \frac{c}{a}$ |
| Sum of squares | $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ |
| Sum of reciprocals | $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}$ |
| Sum of cubes | $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$ |
| Square of difference | $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$ |
| Quadratic from roots | $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$ |

## 下一步

Now that you have mastered functions of quadratic roots, you are ready to move on to more advanced quadratic function topics in Edexcel IGCSE Further Pure Math 4PM1. This skill is foundational for many algebra topics you will encounter if you progress to A Level Further Mathematics, but for your 4PM1 exam, you should practice applying these identities to a range of exam-style questions, including multi-step problems that combine this topic with other algebra skills. Make sure you memorize all the core identities covered, as they are not provided on your formula sheet, and practice forming quadratics with different transformed root types to build confidence for your exam.

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