📚 Topic 159: Completing the Square in Quadratic Equations | 主题159：用配方法解二次方程
Completing the square is a powerful algebraic technique used to solve quadratic equations, find the turning point of a parabola, and derive the quadratic formula. It is a key skill in the Edexcel IGCSE Mathematics syllabus and appears in both the Algebra and Graphs sections.
配方法是一种强大的代数技巧，用于解二次方程、寻找抛物线的顶点，以及推导求根公式。它是爱德思 IGCSE 数学大纲中的关键技能，在代数和函数图像部分均有涉及。
1. Understanding Quadratic Equations | 理解二次方程
A quadratic equation is a polynomial equation of degree 2, generally written in the form:
二次方程是次数为 2 的多项式方程，一般形式为：
ax² + bx + c = 0, 其中 a ≠ 0
Here, a is the coefficient of x², b is the coefficient of x, and c is a constant. The solutions to the equation are the values of x that satisfy it. Quadratic equations can be solved by factorising, using the quadratic formula, or by completing the square.
这里，a 是 x² 的系数，b 是 x 的系数，c 是常数项。方程的解是满足等式的 x 值。二次方程可以通过因式分解、使用求根公式或配方法来求解。
2. What is Completing the Square? | 什么是配方法？
Completing the square rewrites a quadratic expression of the form x² + bx (or ax² + bx) as a perfect square plus or minus a constant. For a quadratic x² + bx, we add (b/2)² and subtract it to keep the expression unchanged:
配方法将形如 x² + bx（或 ax² + bx）的二次表达式改写为完全平方加上或减去一个常数。对于二次表达式 x² + bx，我们加上 (b/2)² 再减去它，以保持表达式不变：
x² + bx = (x + b/2)² − (b/2)²
For example, x² + 6x = (x + 3)² − 9. This method is especially useful when the quadratic cannot be easily factorised.
例如，x² + 6x = (x + 3)² − 9。当二次式不易因式分解时，这种方法尤其有用。
3. Step-by-Step Method | 逐步方法
To complete the square for a monic quadratic (cofficient of x² is 1), follow these steps:
对于首项系数为 1 的二次式（x² 的系数为 1），按以下步骤操作：
- Write the first two terms: x² + bx
- Take half of b: b/2
- Add and subtract (b/2)² inside the expression
- Rewrite as (x + b/2)² − (b/2)²
- 写出前两项：x² + bx
- 取 b 的一半：b/2
- 在表达式中加上并减去 (b/2)²
- 改写为 (x + b/2)² − (b/2)²
When the quadratic has a leading coefficient a ≠ 1, factor out a from the x² and x terms first, then complete the square inside the bracket.
当首项系数 a ≠ 1 时，先从 x² 和 x 项中提取 a，然后在括号内进行配方。
4. Solving Quadratic Equations by Completing the Square | 用配方法解二次方程
To solve a quadratic equation using completing the square, first rewrite the equation in the form (x + p)² = q. Then take the square root of both sides, remembering to include both the positive and negative roots.
要使用配方法解二次方程，首先将方程改写为 (x + p)² = q 的形式。然后对两边取平方根，注意同时取正负根。
Example: Solve x² + 4x – 12 = 0.
示例：解方程 x² + 4x – 12 = 0。
x² + 4x = 12
Complete the square on the left: x² + 4x + 4 – 4 = 12, so:
在左边配方：x² + 4x + 4 – 4 = 12，所以：
(x + 2)² = 16
Taking square roots: x + 2 = ±4, hence x = 2 or x = –6.
取平方根：x + 2 = ±4，因此 x = 2 或 x = –6。
5. Finding Turning Points of Parabolas | 寻找抛物线的顶点
The graph of a quadratic function y = ax² + bx + c is a parabola. Completing the square allows us to write y = a(x + p)² + k, where the turning point (vertex) is at (–p, k). If a is positive, the turning point is a minimum; if a is negative, it is a maximum.
二次函数 y = ax² + bx + c 的图像是一条抛物线。通过配方，我们可以将其写成 y = a(x + p)² + k 的形式，其中顶点坐标为 (–p, k)。若 a 为正，则为最小值点；若 a 为负，则为最大值点。
Example: Find the coordinates of the turning point of y = x² – 6x + 5.
示例：求 y = x² – 6x + 5 的顶点坐标。
y = (x – 3)² – 9 + 5 = (x – 3)² – 4
Thus the turning point is (3, –4).
因此顶点坐标为 (3, –4)。
6. Deriving the Quadratic Formula | 推导求根公式
The quadratic formula x = (–b ± √(b² – 4ac)) / (2a) can be derived by completing the square on the general quadratic ax² + bx + c = 0.
求根公式 x = (–b ± √(b² – 4ac)) / (2a) 可以通过对一般二次方程 ax² + bx + c = 0 配方来推导。
Divide by a: x² + (b/a)x + c/a = 0
两边除以 a：x² + (b/a)x + c/a = 0
Complete the square: (x + b/(2a))² – (b/(2a))² + c/a = 0
配方：(x + b/(2a))² – (b/(2a))² + c/a = 0
Rearrange to solve for x, yielding the famous formula. This derivation is a common examination question in IGCSE.
重新整理求解 x，得到著名的公式。这一推导过程是 IGCSE 考试中常见的题型。
7. Worked Examples | 例题详解
Let’s explore several examples of completing the square with increasing difficulty.
让我们通过几个难度递增的例题来熟悉配方法。
Example 1: Express x² –
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