Mastering Quadratic Equations | 掌握一元二次方程

📚 Mastering Quadratic Equations | 掌握一元二次方程

Quadratic equations appear in nearly every IGCSE Mathematics paper, whether as pure algebra, graphing, or problem-solving questions. Mastering them unlocks a large chunk of marks across Paper 2 and Paper 4. In this guide, we break down every method you need, with clear steps and exam-style examples.

一元二次方程几乎出现在每一份 IGCSE 数学试卷中，无论是纯代数、函数图像还是应用题。掌握这一专题，能帮助你在 Paper 2 和 Paper 4 中拿下大量分数。本指南将逐一拆解所有必备解法，并配有清晰的步骤和考试风格例题。

1. The Standard Form | 标准形式

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. The highest power of the unknown x is 2, which is why these equations are called ‘quadratic’.

一元二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为实数，且 a ≠ 0。未知数 x 的最高次数为 2，因此这些方程被称为”二次”方程。

For example, the equation 2x² – 5x + 3 = 0 is quadratic, with a = 2, b = -5 and c = 3. Note that if a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

例如，方程 2x² – 5x + 3 = 0 是二次方程，其中 a = 2，b = -5，c = 3。注意若 a = 0，方程就退化为一次方程，因此 a ≠ 0 这一条件是至关重要的。

ax² + bx + c = 0, where a ≠ 0

Before solving, always rearrange the equation so that one side is zero. This is the single most important preparation step, and it prevents many sign errors.

在求解之前，务必把方程整理成一边为零的形式。这是最重要的准备步骤，能避免大量符号错误。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer roots. The idea is to write ax² + bx + c as a product of two binomials, then set each factor to zero.

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。其核心思想是将 ax² + bx + c 写成两个二项式的乘积，然后令每个因式分别为零。

For a monic quadratic x²

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