IGCSE Edexcel 数学：配方法解二次方程完全指南 / Solving Quadratic Equations by Completing the Square

引言：为什么”配方法”如此重要？ / Introduction: Why Is Completing the Square So Important?

在 IGCSE Edexcel 数学课程中，解二次方程是代数部分的核心技能之一。你可能已经学会了因式分解法和二次公式法，但还有一种方法既优雅又强大——配方法（Completing the Square）。它不仅是考试中的高频考点（通常出现在 Paper 2 和 Paper 4 中），更是理解二次函数图像和推导二次公式的基础。本文将带你从零开始，系统掌握配方法的每一步。

In the IGCSE Edexcel Mathematics syllabus, solving quadratic equations is one of the core algebraic skills. You have likely learned factorisation and the quadratic formula, but there is another method that is both elegant and powerful — completing the square. It is not only a frequently tested topic (often appearing in Paper 2 and Paper 4) but also the foundation for understanding the graph of quadratic functions and deriving the quadratic formula itself. This article will guide you step by step, from the basics to full mastery.

什么是”配方法”？/ What Is Completing the Square?

简单来说，配方法就是把一个二次三项式 ax² + bx + c 改写为 a(x + p)² + q 的形式。这个”完全平方”的形式让我们能够直接读出抛物线的顶点坐标，并且可以轻松解出方程的根。为什么叫”配方”？因为我们通过加减一个恰当的常数，把不完全的平方表达式”补全”成一个完全平方。

Simply put, completing the square means rewriting a quadratic expression ax² + bx + c into the form a(x + p)² + q. This “completed square” form allows us to directly read off the coordinates of the parabola’s vertex and easily solve for the roots of the equation. Why is it called “completing” the square? Because we add and subtract an appropriate constant to “complete” an incomplete square expression into a perfect square.

核心公式与推导 / The Core Formula and Derivation

对于形如 x² + bx + c 的二次式，配方法的核心操作是：取 x 项系数 b 的一半，平方它，然后同时加上和减去这个值。即：x² + bx = (x + b/2)² − (b/2)²。将这个结果代回原式，即可得到完全平方形式。

For a quadratic expression of the form x² + bx + c, the core operation of completing the square is: take half of the coefficient of x (which is b), square it, then simultaneously add and subtract this value. That is: x² + bx = (x + b/2)² − (b/2)². Substituting this back into the original expression gives the completed square form.

当 x² 的系数不为 1 时（即 ax² + bx + c 且 a ≠ 1），我们需要先将 a 提取出来：ax² + bx + c = a[x² + (b/a)x] + c，然后对括号内的部分进行配方。这是 IGCSE 考试中常见的”升级版”考法。

When the coefficient of x² is not 1 (i.e., ax² + bx + c with a ≠ 1), we must first factor out a: ax² + bx + c = a[x² + (b/a)x] + c, then complete the square inside the brackets. This is a common “advanced” variation in IGCSE exams.

标准步骤：六步法 / Standard Steps: The Six-Step Method

第一步：确保 x² 的系数为 1。 如果 x² 前面有系数（如 2x²、3x² 等），先将该系数从 x² 和 x 项中提取出来。

Step 1: Ensure the coefficient of x² is 1. If there is a coefficient in front of x² (e.g., 2x², 3x²), factor it out from the x² and x terms first.

第二步：将 x 项系数的一半平方。 取 x 的系数，除以 2，然后平方。

Step 2: Square half the coefficient of x. Take the coefficient of x, divide it by 2, then square it.

第三步：同时加减这个平方值。 在表达式中加上再减去这个值，保持等值不变。

Step 3: Add and subtract this squared value. Insert both + and − of this value into the expression, keeping it equivalent.

第四步：将前三项写成完全平方。 x² + bx + (b/2)² 可以写成 (x + b/2)²。

Step 4: Write the first three terms as a perfect square. x² + bx + (b/2)² can be written as (x + b/2)².

第五步：合并常数项。 将剩余的常数项合并化简。

Step 5: Combine the constant terms. Simplify by combining the remaining constant terms.

第六步（解方程时）：移项并开平方。 如果解方程，将完全平方部分移到等号一边，然后两边开平方，记得加上正负号。

Step 6 (when solving equations): Isolate the square and take square roots. If solving an equation, isolate the squared term on one side, then take the square root of both sides, remembering the ± sign.

范例一：基础题 / Example 1: Basic Question

题目：用配方法解方程 x² + 6x + 5 = 0。

Question: Solve x² + 6x + 5 = 0 by completing the square.

解答：
x² + 6x + 5 = 0
x² + 6x = −5   [将常数项移至右边 / Move constant to RHS]
x² + 6x + 9 = −5 + 9   [加上 (6/2)² = 9 / Add (6/2)² = 9]
(x + 3)² = 4   [左边写成完全平方 / Write LHS as perfect square]
x + 3 = ±√4   [两边开平方 / Take square root of both sides]
x + 3 = ±2
x = −3 ± 2
x = −1 或 x = −5
答案：x = −1, x = −5

Solution:
x² + 6x + 5 = 0
x² + 6x = −5   [Move constant to RHS]
x² + 6x + 9 = −5 + 9   [Add (6/2)² = 9]
(x + 3)² = 4   [Write LHS as perfect square]
x + 3 = ±√4   [Take square root of both sides]
x + 3 = ±2
x = −3 ± 2
x = −1 or x = −5
Answer: x = −1, x = −5

范例二：x² 系数不为 1 / Example 2: Coefficient of x² ≠ 1

题目：用配方法解方程 2x² − 8x + 3 = 0。结果保留根号形式。

Question: Solve 2x² − 8x + 3 = 0 by completing the square. Leave your answer in surd form.

解答：
2x² − 8x + 3 = 0
2(x² − 4x) + 3 = 0   [提取 x² 的系数 2 / Factor out 2]
2(x² − 4x) = −3
x² − 4x = −3/2   [两边除以 2 / Divide both sides by 2]
x² − 4x + 4 = −3/2 + 4   [加上 (−4/2)² = 4 / Add (−4/2)² = 4]
(x − 2)² = 5/2   [−3/2 + 4 = −3/2 + 8/2 = 5/2]
x − 2 = ±√(5/2)
x = 2 ± √(5/2)   或写作 / or: x = 2 ± √10/2
答案：x = 2 ± √(5/2)

Solution:
2x² − 8x + 3 = 0
2(x² − 4x) + 3 = 0   [Factor out 2]
2(x² − 4x) = −3
x² − 4x = −3/2   [Divide both sides by 2]
x² − 4x + 4 = −3/2 + 4   [Add (−4/2)² = 4]
(x − 2)² = 5/2   [−3/2 + 4 = 5/2]
x − 2 = ±√(5/2)
x = 2 ± √(5/2)   or: x = 2 ± √10/2
Answer: x = 2 ± √(5/2)

配方法的几何意义 / The Geometric Meaning of Completing the Square

配方法并非只是代数技巧——它有直观的几何解释。考虑 x² + 6x，这可以看作一个边长为 x 的正方形加上一个 6 × x 的矩形。将这个矩形分成两个 3 × x 的窄矩形，分别放在正方形的右侧和下方，会形成一个缺角的大正方形——缺的正是一个 3 × 3 的小正方形。加上这个缺角（即 +9），就得到了一个边长为 (x + 3) 的完整正方形。这就是”配平方”的由来。

Completing the square is not just an algebraic trick — it has an intuitive geometric interpretation. Consider x² + 6x, which can be visualised as a square of side x plus a 6 × x rectangle. Splitting this rectangle into two 3 × x strips and placing them on the right and bottom of the square creates an incomplete larger square — the missing piece is a 3 × 3 small square. Adding this missing corner (+9) completes a perfect square of side (x + 3). This is literally where the name comes from.

配方法的两大核心应用 / Two Core Applications of Completing the Square

1. 求二次函数的顶点 / Finding the Vertex of a Quadratic Function

将二次函数写成 y = a(x + p)² + q 的形式后，顶点坐标即为 (−p, q)。例如，y = (x + 3)² − 4 的顶点是 (−3, −4)。如果 a > 0，抛物线开口向上，顶点是最小值点；如果 a < 0，开口向下，顶点是最大值点。这在 IGCSE 的应用题中非常实用——比如求抛物线的最大高度或最小成本。

Once a quadratic function is written as y = a(x + p)² + q, the coordinates of the vertex are simply (−p, q). For example, y = (x + 3)² − 4 has its vertex at (−3, −4). If a > 0, the parabola opens upwards and the vertex is a minimum point; if a < 0, it opens downwards and the vertex is a maximum point. This is extremely useful in IGCSE application problems — such as finding the maximum height of a projectile or the minimum cost.

2. 推导二次公式 / Deriving the Quadratic Formula

你每天使用的二次公式 x = [−b ± √(b² − 4ac)] / 2a 正是通过配方法从一般形式 ax² + bx + c = 0 推导出来的。理解了配方法，你就不会再”死记”二次公式——你可以自己推导它。

The quadratic formula you use every day — x = [−b ± √(b² − 4ac)] / 2a — is derived directly from completing the square on the general form ax² + bx + c = 0. Once you understand completing the square, you no longer need to “blindly memorise” the quadratic formula — you can derive it yourself.

常见错误与避坑指南 / Common Mistakes and How to Avoid Them

错误一：忘记处理 x² 的系数。 很多学生在面对 3x² + 12x + 7 时，直接将 12 除以 2 再平方，得到 +36 加上去——这是错误的。必须先提取 3。

Mistake 1: Forgetting to handle the coefficient of x². Many students, when faced with 3x² + 12x + 7, directly halve 12 and square it, adding +36 — this is wrong. You must factor out the 3 first.

错误二：忘记平衡等式。 在方程式两边同时加一个数时，左边加了，右边忘了加——这是最常见的失分原因之一。

Mistake 2: Forgetting to balance the equation. When adding a number to both sides of an equation, adding it to the left but forgetting the right — this is one of the most common causes of lost marks.

错误三：开平方时忘记 ± 号。 方程 x² = 9 的解是 x = ±3，不是 x = 3。二次方程通常有两个解（除非判别式为零）。

Mistake 3: Forgetting the ± sign when taking square roots. The equation x² = 9 has solutions x = ±3, not just x = 3. Quadratic equations typically have two solutions (unless the discriminant is zero).

错误四：符号搞错。 (x + b/2)² 展开后是 x² + bx + (b/2)²，而 (x − b/2)² 展开后是 x² − bx + (b/2)²。中间项的符号取决于括号内的符号。

Mistake 4: Getting signs wrong. (x + b/2)² expands to x² + bx + (b/2)², while (x − b/2)² expands to x² − bx + (b/2)². The sign of the middle term depends on the sign inside the bracket.

IGCSE Edexcel 考试技巧 / IGCSE Edexcel Exam Tips

1. 看清题目要求：如果题目明确要求 “by completing the square”，即使你能用因式分解或二次公式快速得到答案，也必须展示配方法的完整过程，否则不给过程分。

1. Read the question carefully: If the question explicitly states “by completing the square”, you must show the full completing-the-square process even if you can get the answer quickly by factorisation or the quadratic formula — otherwise you lose method marks.

2. 保留根号：Paper 2（可以使用计算器）中通常需要精确值；Paper 4 中通常要求保留根号形式（surd form）。

2. Leave answers in surd form: Paper 2 (calculator) typically requires exact values; Paper 4 often requires answers in surd form.

3. 验算方法：将你的答案代入原方程，或者展开你的完全平方形式验证是否等于原式。

3. Check your answer: Substitute your solutions back into the original equation, or expand your completed square form to verify it equals the original expression.

4. 分数分配：这类题目通常值 4-6 分。展示清晰的步骤，即使最后答案不对，也能拿到大部分过程分。

4. Mark allocation: These questions are typically worth 4-6 marks. Show clear working — even if your final answer is wrong, you can earn most of the method marks.

进阶练习 / Practice Questions

试试以下练习题，检验你的掌握程度：

Try these practice questions to test your understanding:

1. 用配方法解 x² − 10x + 21 = 0 (Solve x² − 10x + 21 = 0 by completing the square)

2. 用配方法解 3x² + 12x − 5 = 0，答案保留根号形式 (Solve 3x² + 12x − 5 = 0 by completing the square, leaving answers in surd form)

3. 将 y = x² − 6x + 14 写成 y = (x + p)² + q 的形式，并指出其最小值和对应的 x 值 (Express y = x² − 6x + 14 in the form y = (x + p)² + q, and state its minimum value and the corresponding x-value)

4. 某抛物线的方程为 y = −2x² + 8x − 5。通过配方法找出其最大点坐标。 (A parabola has equation y = −2x² + 8x − 5. By completing the square, find the coordinates of its maximum point.)

（答案见文末 / Answers at the end of the article）

总结 / Summary

配方法是 IGCSE Edexcel 数学中不可或缺的技能。它不仅帮助你解二次方程，更是连接代数与几何的桥梁——让你理解抛物线的对称性、顶点位置和开口方向。掌握本文中的六步法和避坑指南，配方法将从”难点”变成你的”得分点”。多加练习，你一定能在考试中游刃有余！

Completing the square is an indispensable skill in IGCSE Edexcel Mathematics. It not only helps you solve quadratic equations but also serves as a bridge between algebra and geometry — allowing you to understand the symmetry, vertex, and direction of parabolas. Master the six-step method and common mistake guide in this article, and completing the square will transform from a “difficult topic” into your “scoring weapon”. With enough practice, you’ll handle it with ease in the exam!

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练习题答案 / Practice Question Answers

1. x² − 10x + 21 = 0 → (x − 5)² − 25 + 21 = 0 → (x − 5)² = 4 → x = 5 ± 2 → x = 7 或 x = 3

2. 3x² + 12x − 5 = 0 → 3(x² + 4x) = 5 → x² + 4x = 5/3 → (x + 2)² = 5/3 + 4 = 17/3 → x = −2 ± √(17/3)

3. y = x² − 6x + 14 = (x − 3)² − 9 + 14 = (x − 3)² + 5，最小值为 5（当 x = 3 时取得）

4. y = −2x² + 8x − 5 = −2(x² − 4x) − 5 = −2[(x − 2)² − 4] − 5 = −2(x − 2)² + 8 − 5 = −2(x − 2)² + 3，最大点为 (2, 3)