Completing the Square | 配方法

📚 Completing the Square | 配方法

Welcome to the TutorHao IGCSE Maths revision series. Today, we are diving into one of the most powerful algebraic techniques in your exam toolkit: Completing the Square. This method allows you to solve quadratic equations that cannot be factorised easily, and it also reveals the turning point of a quadratic graph in a single step. By the end of this revision guide, you will be able to rewrite any quadratic expression in completed-square form, solve complex equations, and sketch accurate parabolic curves.

欢迎来到 TutorHao IGCSE 数学复习系列。今天，我们将深入探讨考试工具箱中最强大的代数技巧之一：配方法（Completing the Square）。这个方法不仅能让你轻松解出那些无法用因式分解法处理的二次方程，还能一步到位地揭示二次函数图像的顶点坐标。学完本复习指南后，你将能够将任意二次表达式改写为完全平方形式，求解复杂方程，并绘制准确的抛物线草图。

1. The Perfect Square Identity | 完全平方式

Before we start completing the square, we must first recognise a perfect square. A perfect square expression is one that can be written as a single binomial squared. The identity you must memorise is: a² + 2ab + b² = (a + b)². For example, x² + 6x + 9 is a perfect square because it equals (x + 3)².

在开始”配方”之前，我们首先要学会识别完全平方式。完全平方式的特征是可以写成一个二项式的平方。你必须牢记的核心恒等式是：a² + 2ab + b² = (a + b)²。例如，x² + 6x + 9 就是一个完全平方式，因为它等于 (x + 3)²。

Let’s look at the pattern. The coefficient of x is 6, and 6 divided by 2 is 3. The constant term is 9, which is 3². This pattern is the secret behind completing the square: we force the expression to fit this perfect square template.

让我们观察一下规律。一次项系数是 6，6 除以 2 等于 3。常数项是 9，恰好是 3²。这就是配方背后的秘密：我们通过变形，让原表达式尽量

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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