📚 Mastering Completing the Square for IGCSE Quadratic Equations | IGCSE 数学：配方法解二次方程
Completing the square is a core algebraic technique that every IGCSE Mathematics student needs to master. It allows you to solve quadratic equations, sketch parabolas accurately, and understand the structure of quadratic expressions. In this revision guide, you will learn the method step by step, work through graded examples, and discover how to avoid the most common mistakes.
配方法是每位 IGCSE 数学学生都必须掌握的核心代数技巧。它能帮助你解二次方程、准确地画出抛物线图像，并理解二次表达式的结构。在本复习指南中，你将逐步学习配方法，练习分级例题，并学会如何避免最常见的错误。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is any equation that can be arranged into the standard form ax² + bx + c = 0, where a, b and c are constants and a must not be zero. The term ax² is the quadratic term, bx is the linear term, and c is the constant term.
二次方程是任何可以整理成标准形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数，且 a 不能为零。ax² 是二次项，bx 是一次项，c 是常数项。
ax² + bx + c = 0, a ≠ 0
Examples include x² − 5x + 6 = 0, 2x² + 3x − 1 = 0, and −x² + 4x = 0. Quadratic equations arise in area problems, projectile motion, and many other IGCSE applications.
常见的例子包括 x² − 5x + 6 = 0、2x² + 3x − 1 = 0 和 −x² + 4x = 0。二次方程出现在面积问题、抛体运动以及许多其他 IGCSE 应用题中。
Because the highest power of x is 2, a quadratic equation can have up to two real solutions. These solutions are also called roots or x-intercepts of the graph y = ax² + bx + c.
由于 x 的最高次数是 2，二次方程最多可以有两个实数解。这些解也称为根或图像 y = ax² + bx + c 的 x 轴截距。
2. Why Do We Complete the Square? | 为什么要配方？
Factorising is quick and efficient when a quadratic has simple integer roots. For example, x² + 5x + 6 factorises as (x + 2)(x + 3). However, many quadratics do not factorise neatly, especially when the roots are irrational or involve surds.
当二次方程有简单的整数根时，因式分解既快又高效。例如，x² + 5x + 6 可以分解为 (x + 2)(x + 3)。然而，许多二次方程并不能整齐地分解，尤其是当根是无理数或包含根号时。
Completing the square is a systematic method that works for every quadratic equation. It also transforms the expression into vertex form, which makes the turning point of the graph immediately visible.
配方法是一种系统化的方法，适用于所有二次方程。它还能将表达式转化为顶点式，使图像的转折点一目了然。
ax² + bx + c → a(x + p)² + q
This form is essential for sketching parabolas, finding maximum or minimum values, and deriving the quadratic formula. IGCSE exam papers often ask for the completed square form directly.
这种形式对于绘制抛物线草图、求最大值或最小值以及推导求根公式都至关重要。IGCSE 试卷经常直接要求写出配平方后的形式。
3. Perfect Square Trinomials | 完全平方三项式
A perfect square trinomial is produced when a binomial is squared. For example, (x + 3)² expands to x² + 6x + 9, and (x − 5)² expands to x² − 10x + 25.
完全平方三项式由二项式平方产生。例如，(x + 3)² 展开为 x² + 6x + 9，(x − 5)² 展开为 x² − 10x + 25。
The general pattern is shown below. The middle term is always twice the product of x and p, and the last term is always p².
一般规律如下所示。中间项始终是 x 与 p 乘积的两倍，最后一项始终是 p²。
(x + p)² = x² + 2px + p²
When completing the square, we look at the coefficient of x, divide it by 2, and then square the result to create the required constant term.
配方时，我们观察 x 的系数，将其除以 2，然后平方，从而得到所需的常数项。
For x² + 10x, half of 10 is 5, and 5² = 25. So adding 25 gives x² + 10x + 25, which is a perfect square: (x + 5)².
对于 x² + 10x，10 的一半是 5，而 5² = 25。因此加上 25 后得到 x² + 10x + 25，这是一个完全平方：(x + 5)²。
4. Step-by-Step Method When a = 1 | 当 a = 1 时的步骤
If the quadratic is x² + bx + c, follow the steps below to complete the square.
如果二次式为 x² + bx + c，请按照以下步骤进行配方。
Step 1: Keep the x² and x terms together, and move the constant c to the other side if you are solving an equation.
步骤一：将 x² 项和 x 项保留在一起；如果你在解方程，则将常数 c 移到等号另一边。
Step 2: Take half of the coefficient of x. This is b/2.
步骤二：取 x 系数的一半，即 b/2。
Step 3: Square this value to obtain (b/2)².
步骤三：将这个值平方，得到 (b/2)²。
Step 4: Add (b/2)² to both sides of the equation, or add and subtract it when rewriting an expression.
步骤四：在方程两边同时加上 (b/2)²；若是改写表达式，则同时加上并减去它。
Step 5: Factor the perfect square trinomial and simplify the constants.
步骤五：将完全平方三项式分解因式，并化简常数部分。
x² + bx + c = (x + b/2)² + c − (b/2)²
These steps look abstract at first, but they become natural with practice. Always check your final expression by expanding it back.
这些步骤起初看起来抽象，但经过练习后就会变得自然。最后一定要通过展开来检查你的结果。
5. Worked Example: x² + 6x + 2 = 0 | 例题：x² + 6x + 2 = 0
Let us solve the equation x² + 6x + 2 = 0 by completing the square. Start by moving the constant term to the right-hand side.
让我们用配方法解方程 x² + 6x + 2 = 0。首先将常数项移到等号右边。
x² + 6x = −2
Take half of 6, which is 3, and square it to get 9. Add 9 to both sides of the equation.
取 6 的一半，即 3，平方后得到 9。在方程两边同时加上 9。
x² + 6x + 9 = −2
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