📚 Solving Quadratic Equations | 解二次方程
Quadratic equations are one of the most heavily tested topics in IGCSE Mathematics. You need to solve them in algebra, interpret them in graphs, and apply them to word problems. This guide explains every method you need and shows you exactly how to write your working.
二次方程是 IGCSE 数学中考查最频繁的内容之一。你需要在代数中求解它们、在图像中理解它们，并将它们应用于文字应用题。本指南讲解你需要的所有方法，并说明如何规范写出解题过程。
1. Quadratic Equations and Their Standard Form | 二次方程及其标准形式
A quadratic equation can be written in the standard form:
ax² + bx + c = 0, where a ≠ 0
Here a, b and c are numbers; x is the unknown. The highest power of x is 2, which is why it is called "quadratic".
这里 a、b 和 c 是常数，x 是未知数。未知数最高次数是 2，所以称为“二次”方程。
For example, 3x² - 2x + 1 = 0 has a = 3, b = -2, c = 1.
例如，3x² - 2x + 1 = 0 中 a = 3，b = -2，c = 1。
2. Expanding to Check Factorisation | 展开与检验因式分解
Before solving, make sure you can expand brackets correctly.
在求解之前，请确保你能正确展开括号。
(x + p)(x + q) = x² + (p + q)x + pq
This identity is the key to checking a factorisation. Expand your factors mentally to confirm they match the original expression.
这个恒等式是检验因式分解的关键。在脑中展开因式，检查是否与原来的表达式一致。
Example: (x + 4)(x - 2) = x² + 4x - 2x - 8 = x² + 2x - 8.
例如：(x + 4)(x - 2) = x² + 4x - 2x - 8 = x² + 2x - 8。
3. Solving by Factorisation | 用因式分解求解
Solving by factorisation uses the zero product property: if AB = 0, then A = 0 or B = 0.
用因式分解求解的依据是零乘积性质：如果 AB = 0，那么 A = 0 或 B = 0。
Follow these steps:
按照以下步骤进行：
- Rearrange into ax² + bx + c = 0. 将方程整理为 ax² + bx + c = 0。
- Factorise the quadratic expression. 对二次表达式进行因式分解。
- Set each bracket equal to zero and solve. 令每个括号等于零并分别求解。
Solve x² - 5x + 6 = 0. Since (x - 2)(x - 3) = 0, the solutions are x = 2 or x = 3.
解 x² - 5x + 6 = 0。因为 (x - 2)(x - 3) = 0，所以解为 x = 2 或 x = 3。
4. The Difference of Two Squares | 平方差公式
The difference of two squares is a special pattern that appears often:
平方差公式是一种常考的特殊模式：
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