Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most frequently tested topics in the IGCSE Mathematics syllabus. Whether you sit for the Core or Extended paper, questions on factorisation, the quadratic formula, completing the square, the discriminant, and graphical interpretation appear year after year. This revision guide covers every essential skill from the basics to exam-level application, with clear examples and common-mistake warnings to help you secure full marks.

二次方程是 IGCSE 数学大纲中最常考的内容之一。无论你参加 Core 还是 Extended 试卷，因式分解、求根公式、配方法、判别式和图像解读的题目年复一年地出现。本复习指南将带你掌握从基础到考试级应用的所有必备技能，配以清晰的例题和常见错误警示，帮助你冲击满分。

1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is any equation that can be written in the general form:

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are real constants. The value a is called the leading coefficient, and it must not be zero; otherwise the equation becomes linear. The highest power of x is 2, which is what makes the equation quadratic.

这里 a、b、c 是实数常数。a 称为首项系数，它不能为零；否则方程就退化为一次方程。未知数 x 的最高次数是 2，这正是二次方程的判别特征。

For example, 3x² − 5x + 2 = 0 is already in standard form with a = 3, b = −5 and c = 2. However, x² = 4x + 7 is not in standard form yet — you must rearrange it to x² − 4x − 7 = 0 before applying any solution method.

例如，3x² − 5x + 2 = 0 已经是标准形式，其中 a = 3，b = −5，c = 2。但 x² = 4x + 7 还不是标准形式——你必须先整理为 x² − 4x − 7 = 0，然后才能使用任何解法。

2. Solving by Factorisation | 用因式分解法求解

Factorisation is the quickest method when the quadratic has simple integer roots. The underlying principle is the zero product property: if p × q = 0, then either p = 0 or q = 0.

当二次方程具有简单的整数根时，因式分解是最快捷的方法。其基本原理是零积性质：若 p × q = 0，则 p = 0 或 q = 0。

Example: Solve x² + 5x + 6 = 0.

例题：解方程 x² + 5x + 6 = 0。

We need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3. Hence:

我们需要找到两个数，它们相乘等于 6，相加等于 5。这两个数就是 2 和 3。因此：

(x + 2)(x + 3) = 0

Now apply the zero product property:

现在应用零积性质：

x + 2 = 0 → x = −2  or  x + 3 = 0 → x = −3

When the coefficient of x² is not 1, the factorisation is slightly harder. For example, 2x² + 7x + 3 = 0. We look for factor pairs of 2 × 3 = 6 that add to 7: those are 1 and 6. We then split the middle term:

当 x² 的系数不为 1 时，因式分解稍微困难一些。例如，2x² + 7x + 3 = 0。我们先找出 2 × 3 = 6 的因数对，使其和为 7：即 1 和 6。然后拆开中间项：

2x² + x + 6x + 3 = 0 → x(2x + 1) + 3(2x + 1) = 0 → (2x + 1)(x + 3) = 0

Therefore x = −½ or x = −3. Always expand your factorised answer mentally to confirm the factors are correct before moving on.

因此 x = −½ 或 x = −3。在继续之前，务必在心中展开因式进行验算，确保因式没有找错。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This technique is essential when factorisation is not possible, and it is

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