📚 Completing the Square for IGCSE Mathematics | IGCSE 数学配方法解二次方程
Completing the square is one of the most important algebraic techniques in IGCSE Mathematics. It allows you to solve quadratic equations, rewrite quadratic expressions in a useful form, and identify the turning point of a quadratic graph.
配方法是 IGCSE 数学中最重要的代数技巧之一。它可以帮助你解二次方程、改写二次表达式，并确定二次函数图像的顶点。
This method is assessed regularly in both the core and extended IGCSE papers. Understanding the steps clearly will give you confidence when handling quadratics in any form.
该方法在 IGCSE 核心卷和扩展卷中经常出现。清楚地理解配方法的步骤，能让你在处理各种二次方程时更有把握。
1. What Is Completing the Square? | 什么是配方法
Completing the square is a technique that rewrites a quadratic expression of the form ax² + bx + c into a new form: a(x + p)² + q.
配方法是一种将形如 ax² + bx + c 的二次表达式改写为 a(x + p)² + q 形式的技巧。
In this new form, the expression inside the bracket, (x + p)², is a perfect square trinomial. This makes the equation much easier to solve or graph.
在这种新形式中，括号内的 (x + p)² 是一个完全平方式。这让方程更容易求解或作图。
The name 'completing the square' comes from the idea of adding a specific number to make a perfect square trinomial.
“配方法”这个名称来源于加上一个特定的数，从而凑成一个完全平方式。
ax² + bx + c = a(x + p)² + q
CSE, you must be able to complete the square both when a = 1 and when a is not equal to 1.
在 IGCSE 考试中，你需要掌握当 a = 1 和 a ≠ 1 时的配方法。
2. The Standard Quadratic Form | 二次方程的标准形式
e written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
每个二次方程都可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。
The coefficient a is the leading coefficient, b is the coefficient of x, and c is the constant term.
系数 a 是二次项系数，b 是 x 的系数，c 是常数项。
Before you begin completing the square, always check that the equation is arranged in this standard order.
在开始配方法之前，务必检查方程是否已经按这个标准顺序排列。
ax² + bx + c = 0
order, rearrange them. For example, 4x + x² − 5 = 0 should be written as x² + 4x − 5 = 0.
如果各项顺序混乱，要先重新排列。例如，4x + x² − 5 = 0 应写为 x² + 4x − 5 = 0。
3. Step 1: Make the Leading Coefficient 1 | 第一步：使二次项系数为 1
If the leading coefficient a is not equal to 1, divide every term of the equation by a. This makes the coefficient of x² equal to 1.
如果二次项系数 a 不等于 1，就将方程中的每一项都除以 a，使 x² 的系数变为 1。
Example: For the equation 2x² + 8x + 6 = 0, divide every term by 2:
例如：对于方程 2x² + 8x + 6 = 0，将每一项除以 2：
2x² + 8x + 6 = 0 → x² + 4x + 3 = 0
When a is already 1, you can skip this step and move directly to moving the constant term.
当 a 已经等于 1 时，可以跳过这一步，直接进行移常数项。
This division step is essential because the standard method only works simply when the coefficient of x² is 1.
这个除法步骤非常重要，因为只有当 x² 的系数为 1 时，标准配方法步骤才会最简便。
4. Step 2: Move the Constant Term | 第二步：移常数项
Rearrange the equation so that the constant term is on the right-hand side. Keep the x² term and the x term on the left.
重新整理方程，使常数项移到右边，将 x² 项和 x 项保留在左边。
From the equation x² + 4x + 3 = 0, subtract 3 from both sides:
对于方程 x² + 4x + 3 = 0，两边同时减去 3：
