📚 IGCSE Maths: Completing the Square for Quadratic Equations | IGCSE 数学：二次方程的配方法
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to real-world problems. One powerful technique for solving any quadratic equation is completing the square. This method transforms a quadratic expression into a perfect square plus or minus a constant, making it easier to solve, sketch, and analyse.
二次方程贯穿 IGCSE 数学课程的代数、函数图像和实际应用。配方法是一种强大的求解二次方程的技术，它把二次表达式转化为完全平方加上或减去一个常数，使求解、画图和性质分析都更加容易。
1. What Is Completing the Square? | 什么是配方法？
Completing the square rewrites a quadratic expression such as x² + bx + c in the form (x + p)² + q, where p and q are constants. The left-hand side becomes a perfect square trinomial, and the remaining term adjusts the value. This is especially useful when a quadratic cannot be factorised easily.
配方法将形如 x² + bx + c 的二次表达式改写为 (x + p)² + q 的形式，其中 p 和 q 是常数。左边变成完全平方三项式，剩下的常数项调节数值。当二次式不容易因式分解时，这一方法尤其有用。
Completing the square has three main advantages:
配方法有三大优势：
- It always works for any quadratic equation.
- It reveals the vertex of a parabola directly.
- It helps derive the quadratic formula.
- 对任意二次方程都适用；
- 直接给出抛物线的顶点；
- 可用于推导求根公式。
2. The Standard Quadratic Form | 二次方程的标准形式
ere a, b and c are constants and a ≠ 0. The coefficient a controls the width and direction of the parabola, b affects the axis position, and c is the y-intercept. Completing the square usually starts by identifying these three values.
一元二次方程的标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。系数 a 决定抛物线的开口方向和宽窄，b 影响对称轴位置，c 是 y 轴截距。配方法通常从识别这三个数值开始。
ax² + bx + c = 0
3. Step-by-Step Method for a = 1 | 当 a = 1 时的配方法步骤
e the constant term to the right-hand side; (2) Take half of the coefficient of x, square it, and add it to both sides; (3) Write the left side as a perfect square; (4) Take square roots and solve for x.
当 a = 1 时，可按以下步骤操作：(1) 把常数项移到右边；(2) 取 x 系数的一半并平方，再加到等号两边；(3) 将左边写成完全平方；(4) 开平方并解出 x。
x² + bx = -c → x² + bx + (b/2)² = -c + (b/2)²
The critical step is adding (b/2)² to both sides; otherwise the equation changes meaning.
关键步骤是在等号两边同时加上 (b/2)²，否则方程的意义会发生改变。
4. Worked Example: x² + 6x + 5 = 0 | 例题：x² + 6x + 5 = 0
or the left side: (x + 3)² = 4. Taking square roots gives x + 3 = ±2, so x = -1 or x = -5.
用配方法解 x² + 6x + 5 = 0。先把常数项移项：x² + 6x = -5。6 的一半是 3，3² = 9。两边加 9：x² + 6x + 9 = 4。左边因式分解为 (x + 3)² = 4。开平方得 x + 3 = ±2，所以 x = -1 或 x = -5。
(x + 3)² = 4 ⇒ x + 3 = ±2 ⇒ x = -1 or -5
Always check by substituting both answers into the original equation.
始终把两个答案代回原方程进行检验。
5. Handling Coefficients When a ≠ 1 | 当 a ≠ 1 时如何处理系数
x - 10 = 0 becomes x² + 4x - 5 = 0 after dividing by 2. Then complete the square as usual.
如果 a ≠ 1，首先将整个方程除以 a，或者从含变量的项中提出 a。例如，2x² + 8x - 10 = 0 除以 2 后变为 x² + 4x - 5 = 0，然后照常配方。
2x² + 8x - 10 = 0 → x² + 4x - 5 = 0 → (x + 2)² = 9
Do not complete the square before removing the leading coefficient; the half-and-square rule only applies to the x coefficient after the x² coefficient is 1.
在提取首项系数之前不要直接配方；“取一半再平方”规则只适用于 x² 系数为 1 时。
6. Using Completing the Square to Derive the Quadratic Formula | 用配方法推导求根公式
a, move c/a to the right, add (b/2a)² to both sides, and simplify. The result is x = [-b ± √b² - 4ac] / (2a).
求根公式正是对一般方程 ax² + bx + c = 0 配方得到的。先除以 a，把 c/a 移到右边，两边加 (b/2a)²，再化简，就得到 x = [-b ± √b² - 4ac] / (2a)。
x = [-b ± √b² - 4ac] / (2a)
This derivation is a common IGCSE exam question, so understanding the algebra behind the formula is valuable.
这种推导是 IGCSE 考试中常见的题型，因此理解公式背后的代数过程很有价值。
7. Finding the Vertex of a Quadratic Graph | 求二次函数图像的顶点
omes y = (x + 3)² - 4, so the vertex is (-3, -4). The line x = -3 is the axis of symmetry.
配方法可将 y = ax² + bx + c 转化为顶点式 y = a(x - h)² + k，其中 (h, k) 为顶点。例如 y = x² + 6x + 5 可化为 y = (x + 3)² - 4，因此顶点是 (-3, -4)，对称轴为 x = -3。
y = x² + 6x + 5 = (x + 3)² - 4
When a > 0 the vertex is a minimum point; when a < 0 it is a maximum point.
当 a > 0 时，顶点是最低点；当 a < 0 时，顶点是最高点。
8. Solving Real-World Problems | 解决实际问题
t in metres after t seconds, completing the square shows the maximum height and when it occurs.
许多 IGCSE 题目涉及最大面积、利润或抛射高度。配方法能快速给出最大值或最小值。例如，若 h = -t² + 6t + 7 表示 t 秒后的高度（米），配方后可显示最大高度及发生时间。
h = -t² + 6t + 7 = -(t - 3)² + 16
The vertex (3, 16) means the maximum height is 16 metres at t = 3 seconds.
顶点 (3, 16) 表示最大高度为 16 米，发生在 t = 3 秒时。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One common mistake is adding (b/2)² to only one side of the equation. Another is forgetting to divide by a first when a ≠ 1. Also, when taking square roots, both positive and negative roots must be considered.
一个常见错误是只在方程的一边加 (b/2)²。另一个是当 a ≠ 1 时忘记先除以 a。此外，开平方时必须考虑正负两个根。
Key reminders:
关键提醒：
- Always add the same value to both sides.
- Check the sign of the constant term after completing the square.
- Write both roots explicitly: x + p = ±√q.
- 始终在两边加相同的值；
- 配方后检查常数项符号；
- 明确写出正负两个根：x + p = ±√q。
10. Exam Tips and Summary | 考试技巧与总结
In the exam, if the question says 'solve by completing the square,' you must show the method rather than just using the formula. Present each line clearly, including the step where you add (b/2)². Use completing the square for finding vertices and deriving the formula. With practice, it becomes a reliable tool for quadratic problems.
考试中如果题目要求“用配方法求解”，就必须展示配方法过程，而不能直接套公式。每一步都要写清楚，包括加上 (b/2)² 的那一步。用配方法求顶点和推导求根公式都很有效。经过练习，它将成为解决二次函数问题的可靠工具。
The table below compares three common methods for solving quadratic equations.
下表比较了三种常用的二次方程求解方法。
| Method | Best for |
|---|---|
| Factorising | Simple integer roots |
| Completing the square | Vertex form and exact surd roots |
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