📚 Completing the Square for IGCSE Mathematics | IGCSE 数学配方法完全指南
Quadratic equations appear throughout the IGCSE Mathematics syllabus, and completing the square is one of the most powerful techniques for solving them and for understanding their graphs. This article guides you through the method step by step, with worked examples, graph connections and exam advice.
二次方程贯穿 IGCSE 数学课程，配方法是求解二次方程和理解其图像最有力的技巧之一。本文将通过分步讲解、例题、图像联系和考试建议，帮助你掌握这一方法。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is any equation that can be written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of x is 2, which means the graph of y = ax² + bx + c is always a parabola.
二次方程是可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。x 的最高次数是 2，因此 y = ax² + bx + c 的图像总是一条抛物线。
For example, x² + 6x + 2 = 0 and 2x² − 5x + 1 = 0 are both quadratic equations. Solving them means finding the values of x that make the equation true. These values are also the x-intercepts of the corresponding parabola.
例如，x² + 6x + 2 = 0 和 2x² − 5x + 1 = 0 都是二次方程。求解就是找出使方程成立的 x 值。这些值同时也是对应抛物线与 x 轴的交点。
2. The Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0
Before starting any method, you should rearrange the equation into standard form. The coefficient a is the number in front of x², b is in front of x, and c is the constant term. This rearrangement makes the roles of a, b and c clear.
在开始任何方法之前，你都应该把方程整理成标准形式。系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。这样整理可以使 a、b、c 的作用一目了然。
ax² + bx + c = 0
For example, the equation 3x = 2 − x² can be rearranged to x² + 3x − 2 = 0. Here a = 1, b = 3 and c = −2. If the x² term appears on the right, move it first so that the standard form is easier to recognise.
例如，方程 3x = 2 − x² 可以整理为 x² + 3x − 2 = 0。这里 a = 1，b = 3，c = −2。如果 x² 项出现在右边，应先将它移到左边，这样更容易识别标准形式。
3. Perfect Square Trinomials | 完全平方三项式
A perfect square trinomial is an expression of the form x² + 2ax + a², which can be written as (x + a)². Recognising these patterns is essential because completing the square turns a general quadratic into this form.
完全平方三项式是形如 x² + 2ax + a² 的表达式，可以写成 (x + a)²。识别这些模式非常重要，因为配方法就是把一般二次式转化为这种形式。
Examples include:
示例包括：
- x² + 10x + 25 = (x + 5)²
- x² − 8x + 16 = (x − 4)²
- x² + 2x + 1 = (x + 1)²
Notice that the constant term is always half the coefficient of x, squared. That is the key fact behind the whole method.
注意，常数项总是 x 项系数一半的平方。这是整个配方法背后的关键事实。
4. Why Completing the Square Works | 配方法为什么有效
Completing the square works by adding and subtracting a cleverly chosen constant to force part of the quadratic into a perfect square. The remaining constant is then adjusted to keep the equation balanced.
配方法的原理是加上并减去一个巧妙选择的常数，使二次式的一部分变成完全平方。剩余的常数随后进行调整，保持方程平衡。
This method is useful because it can solve any quadratic equation, even when factorisation is difficult or impossible. It also reveals the vertex of the parabola directly, which is why it is tested frequently in IGCSE.
这种方法很有用，因为它可以求解任何二次方程，即使因式分解困难或无法进行。它还能直接揭示抛物线的顶点，因此 IGCSE 考试中经常考查。
5. Step-by-Step Method for a = 1 | a = 1 时的分步步骤
When the coefficient of x² is 1, follow these steps carefully:
当 x² 的系数为 1 时，请仔细按照以下步骤操作：
Step 1: Start with x² + bx + c = 0. Move the constant term c to the right-hand side by subtracting c from both sides.
步骤 1：从 x² + bx + c = 0 开始，将常数项 c 移到等号右边，即两边同时减去 c。
Step 2: Add (b/2)² to both sides. This makes the left side a perfect square trinomial.
步骤 2：两边同时加上 (b/2)²。这使得左边成为完全平方三项式。
Step 3: Write the left side as (x + b/2)² and simplify the right side by combining any fractions or integers.
步骤 3：将左边写成 (x + b/2)²，并通过合并分数或整数来化简右边。
Step 4: Take the square root of both sides, remembering the ± sign on the right-hand side.
步骤 4：两边开平方，记得在右边加上 ± 号。
Step 5: Solve for x by isolating the variable. You should obtain two solutions unless the right side is zero, in which case there is one repeated solution.
步骤 5：通过分离变量解出 x。除非右边为零，否则应得到两个解；右边为零时则有一个重根。
6. Worked Example: x² + 6x + 2 = 0 | 例题：x² + 6x + 2 = 0
Let us solve x² + 6x + 2 = 0 by completing the square.
让我们用配方法解方程 x² + 6x + 2 = 0。
First, move the constant to the right:
首先，把常数移到右边：
x² + 6x = −2
The coefficient b is 6, so half of b is 3 and (b/2)² is 9. Add 9 to both sides:
系数 b 是 6，所以 b 的一半是 3，(b/2)² 是 9。两边同时加 9：
x² + 6x + 9 = 7
The left side is now a perfect square:
左边现在是完全平方：
(x + 3)² = 7
Take the square root of both sides:
两边开平方：
x + 3 = ±√7
Finally, subtract 3 from both sides to get the two solutions:
最后，两边减去 3，得到两个解：
x = −3 + √7 or x = −3 − √7
These are the exact solutions. You can leave answers in surd form unless a decimal approximation is asked for.
这些是精确解。除非题目要求小数近似值，否则答案可以保留根号形式。
7. Handling a ≠ 1 | 当 a ≠ 1 时如何处理
If the coefficient of x² is not 1, first make the leading coefficient equal to 1. The safest approach is to divide every term by a, then complete the square as usual. This may create fractions, but the method is identical.
如果 x² 的系数不是 1，首先要使首项系数等于 1。最稳妥的方法是把每一项都除以 a，然后照常配方法。这可能会产生分数
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