Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are a core part of the IGCSE Mathematics syllabus. They appear in algebra, coordinate geometry, and many problem-solving questions. Understanding when and how to apply each method — factorisation, the quadratic formula, completing the square, and graphical methods — is essential for exam success.

一元二次方程是 IGCSE 数学课程的核心内容，出现在代数、坐标几何以及许多应用题中。理解并灵活运用因式分解法、求根公式法、配方法和图像法，是取得考试成功的关键。

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

A quadratic equation is any equation that can be rearranged into the standard form, where a, b and c are constants and a is not zero:

一元二次方程是指可以化成如下标准形式的方程，其中 a、b、c 为常数，且 a 不等于零：

ax² + bx + c = 0, a ≠ 0

For example, x² − 5x + 6 = 0 has a = 1, b = −5, c = 6. The equation 3x² = 12 can be rewritten as 3x² − 12 = 0, so it is also quadratic. However, an equation such as x³ − 4x = 0 is cubic, not quadratic.

例如，x² − 5x + 6 = 0 中 a = 1，b = −5，c = 6。方程 3x² = 12 可改写为 3x² − 12 = 0，因此它也属于二次方程。而 x³ − 4x = 0 是三次方程，不是二次方程。

2. Expanding and Factorising Quick Revision | 展开与因式分解快速复习

Solving quadratics by factorisation relies on the reverse of expanding double brackets. Recall that:

用因式分解法解二次方程，本质上是展开双括号的逆运算。回顾公式：

(x + p)(x +

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