Solving Quadratic Equations | 一元二次方程完全攻略

📚 Solving Quadratic Equations | 一元二次方程完全攻略

Quadratic equations appear in almost every IGCSE Mathematics paper, whether as direct solving questions, word problems, or graph-sketching tasks. Mastering this topic is essential for achieving a top grade, and the good news is that every method you need can be learned with confidence.

一元二次方程几乎出现在每一份 IGCSE 数学试卷中，无论是直接解方程、实际应用题，还是函数图像作图题。掌握这一知识点是冲击高分的关键，而且好消息是：考试所需的所有解法都可以扎实学会。

1. The Standard Form of a Quadratic Equation | 一元二次方程的标准形式

An equation is quadratic if it can be written in the general form:

如果一个方程可以写成如下一般形式，它就是一元二次方程：

ax² + bx + c = 0, where a ≠ 0

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The value of a must not be zero; otherwise the equation becomes linear and loses its quadratic nature.

其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。要注意 a 不能为 0，否则方程会退化成一次方程，不再具备二次特征。

For example, 3x² − 2x + 5 = 0 is quadratic, with a = 3, b = −2 and c = 5. However, the equation x² − 2x − 1 = 2x + 4 is not yet in standard form; you must first rearrange it to x² − 4x − 5 = 0 before choosing a method.

例如，3x² − 2x + 5 = 0 是标准的一元二次方程，其中 a = 3，b = −2，c = 5。而方程 x² − 2x − 1 = 2x + 4 尚未写成标准形式，需要先整理为 x² − 4x − 5 = 0，才能选择合适的解法。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer factors. The key principle is the zero product property: if P × Q = 0, then at least one of P or Q must equal zero.

当二次式含有简单的整数因式时，因式分解是最快的方法。其核心依据是”零积性质”：若 P × Q = 0，则 P 与 Q 中至少有一个等于 0。

Example: solve x² − 5x + 6 = 0. We need two numbers that multiply to 6 and add to −5; those numbers are −2 and −3, so the equation becomes (x − 2)(x − 3) = 0.

例题：解 x² − 5x + 6 = 0。我们需要找两个数，相乘得 6 且相加得 −5；这两个数是 −2 和 −3，于是方程化为 (x − 2)(x − 3) = 0。

(x − 2)(x − 3) = 0, so x = 2 or x = 3

Special cases are common in exams. If b = 0, use the difference of two squares: x² − 9 = 0 gives (x − 3)(x + 3) = 0, so x = ±3. If c =

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