Completing the Square | 配方法

📚 Completing the Square | 配方法

Completing the square is one of the most powerful algebraic techniques in the Edexcel IGCSE Mathematics syllabus. It transforms a quadratic expression into a perfect square plus a constant, which unlocks solutions to equations, reveals turning points of curves, and helps us sketch parabolas with confidence. This article guides you through the method step by step, with worked examples and exam-style practice.

配方法是 Edexcel IGCSE 数学大纲中最实用的代数技巧之一。它能把二次表达式转化为”完全平方加常数”的形式，从而帮助我们解方程、确定曲线的极值点，并轻松绘制抛物线图像。本文将结合例题和考试题型，一步一步带你掌握这一方法。

1. The Core Idea | 核心思想

Consider a quadratic expression of the form x² + bx. We want to write it as a perfect square plus or minus a constant. Recall the standard expansion:

考虑形如 x² + bx 的二次表达式。我们希望把它写作”完全平方加/减常数”的形式。回顾标准展开式：

(x + p)² = x² + 2px + p²

To make the middle term equal bx, we set 2p = b, so p = b/2. Substituting gives:

为了使中间项等于 bx，我们令 2p = b，即 p = b/2。代入得：

(x + b/2)² = x² + bx + (b/2)²

Subtract the extra (b/2)² from both sides to recover the original expression:

两边同时减去多余的 (b/2)²，就能还原原表达式：

x² + bx = (x + b/2)² − (b/2)²

This is the essence of completing the square: we “complete” a perfect square and then correct the constant term.

这就是配方法的本质：我们先”补全”一个完全平方，然后修正常数项。

2. Simple Worked Examples | 基础例题

Let us apply the formula to a few simple expressions. The key is always to halve the coefficient of x and square it.

我们把公式应用到几个简单表达式中。关键步骤永远是：将 x 的系数减半，然后平方。

Expression	b	b/2	Completed Square Form
x² + 8x	8	4	(x + 4)² − 16
x² − 10x	−10	−5	(x − 5)² − 25
x² + 6x + 5	6	3	(x + 3)² − 4

Check the last example by expanding: (x + 3)² − 4 = x² + 6x + 9 − 4 = x² + 6x + 5, which is correct.

验证最后一个例子：展开 (x + 3)² − 4 = x² + 6x + 9 − 4 = x² + 6x + 5，结果正确。

3. General Quadratic: ax² + bx + c | 一般二次式：ax² + bx + c

When the coefficient of x² is not 1, factor it out from the x² term and the x term first. Do not factor it out of the constant term.

当 x² 的系数不为 1 时，先从 x² 项和 x 项中提出该系数，但不要把它从常数项中提出来。

ax² + bx + c = a[x² + (b/a)x] + c

Now complete the square inside the brackets, then multiply the bracket back out and simplify.

然后在方括号内配方，再将方括号项乘回去并化简。

Example | 例题: Express 2x² + 8x + 3 in the form a(x + p)² + q.

例：将 2x² + 8x + 3 化为 a(x + p)² + q 的形式。

2x² + 8x + 3 = 2[x² + 4x] + 3

= 2[(x + 2)² − 4] + 3

= 2(x + 2)² − 8 + 3

= 2(x + 2)² − 5

Expanding the final answer gives 2(x² + 4x + 4) − 5 = 2x² + 8x + 8 − 5 = 2x² + 8x + 3, which confirms the result.

展开最终答案：2(x² + 4x + 4) − 5 = 2x² + 8x + 8 − 5 = 2x² + 8x + 3，验证结果正确。

4. Solving Quadratic Equations | 解一元二次方程

Completing the square provides a reliable method for solving any quadratic equation, even when the quadratic does not factorise.

配方法为解任何一元二次方程都提供了可靠的思路，即使该二次式无法因式分解。

Example 1 | 例 1: Solve x² + 6x − 7 = 0.

例 1：解方程 x² + 6x − 7 = 0。

Move the constant to the right-hand side:

将常数项移到等号右边：

x² + 6x = 7

Complete the square on the left:

在左边配方：

(x + 3)² − 9 = 7

(x + 3)² = 16

Take the square root of both sides, remembering the ± sign:

两边开平方，注意正负号：

x + 3 = ±4

x = 1 or x = −7

Example 2 | 例 2: Solve x² − 8x + 5 = 0, leaving your answer in surd form.

例 2：解方程 x² − 8x + 5 = 0，结果保留根号形式。

x² − 8x = −5

(x − 4)² − 16 = −5

(x − 4)² = 11

x = 4 ± √11

5. Finding Turning Points | 求极值点

Once a quadratic is written as y = (x + p)² + q, it is easy to locate its vertex. Since (x + p)² is always greater than or equal to zero, the smallest value of y is q, which occurs when x = −p. Thus the turning point is (−p, q).

把二次式写成 y = (x + p)² + q 之后，就可以轻松找到顶点。(x + p)² 永远大于等于零，所以 y 的最小值为 q，此时 x = −p。因此极值点为 (−p, q)。

If the leading coefficient is negative, the parabola opens downwards and the turning point is a maximum.

若首项系数为负数，抛物线开口向下，此时极值点为最大值点。

Example | 例题: Find the coordinates of the turning point of y = x² − 6x + 11, and state whether it is a maximum or minimum.

例：求 y = x² − 6x + 11 的极值点坐标，并说明它是最大值还是最小值。

y = (x − 3)² − 9 + 11 = (x − 3)² + 2

Hence the vertex is at (3, 2), and since the coefficient of (x − 3)² is positive, this is a minimum point.

因此顶点为 (3, 2)，而 (x − 3)² 的系数为正，所以这是最小值点。

6. Sketching Quadratic Graphs | 绘制二次函数图像

Completing the square gives us three key features for sketching a parabola: the vertex, the axis of symmetry, and the y-intercept.

配方法可以告诉我们画抛物线所需的三个关键特征：顶点、对称轴和 y 轴截距。

Follow these steps:

按照以下步骤进行：

Complete the square to find the vertex (−p, q).

配方求顶点 (−p, q)。
The axis of symmetry is the vertical line x = −p.

对称轴是竖直直线 x = −p。
Find the y-intercept by setting x = 0.

令 x = 0 求出 y 轴截距。
Find roots by solving the equation, if the parabola crosses the x-axis.

若抛物线与 x 轴相交，解方程求根。
Sketch the smooth U-shaped curve using these points.

用这些点画光滑的 U 形曲线。

Example | 例题: Sketch y = x² − 4x + 3.

例：画出 y = x² − 4x + 3 的图像。

y = (x − 2)² − 4 + 3 = (x − 2)² − 1

Vertex: (2, −1). Axis of symmetry: x = 2. y-intercept: 3. Roots: solve (x − 2)² = 1, giving x = 1 or x = 3.

顶点为 (2, −1)；对称轴为 x = 2；y 轴截距为 3；根：解 (x − 2)² = 1，得 x = 1 或 x = 3。

7. Deriving the Quadratic Formula | 推导求根公式

Completing the square is the method used to derive the general quadratic formula from ax² + bx + c = 0. This derivation is an excellent test of your algebraic manipulation.

从 ax² + bx + c = 0 出发，用配方法可以推导通用求根公式。这一推导过程是对代数运算能力的绝佳考验。

a[x² + (b/a)x] + c = 0

a[(x + b/2a)² − (b/2a)²] + c = 0

a(x + b/2a)² − b²/4a + c = 0

(x + b/2a)² = (b² − 4ac) / 4a²

Taking square roots gives the familiar formula:

两边开平方，得到我们熟悉的公式：

x = (−b ± √(b² − 4ac)) / 2a

You will not normally need to repeat this derivation in an exam, but understanding it helps you remember the formula and know when the discriminant b² − 4ac matters.

考试中通常不需要你重复这一推导过程，但理解它有助于记忆公式，并理解判别式 b² − 4ac 的意义。

8. Exam-Style Practice | 考试题型练习

Try these questions, which are typical of Edexcel IGCSE papers. Attempt each one fully before reading the solution.

以下题目是 Edexcel IGCSE 试题的典型代表。请先完整作答，再对照解答。

Question 1 | 题 1: Express 2x² − 12x + 5 in the form 2(x − p)² + q, where p and q are constants.

题 1：将 2x² − 12x + 5 化为 2(x − p)² + q 的形式，其中 p、q 为常数。

Solution | 解:

2[x² − 6x] + 5 = 2[(x − 3)² − 9] + 5 = 2(x − 3)² − 13

So p = 3 and q = −13.

所以 p = 3，q = −13。

Question 2 | 题 2: Find

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Completing the Square | 配方法