Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in almost every exam paper, either as standalone questions or as part of problem-solving tasks. This guide will walk you through every method you need to know, from factorisation to the quadratic formula, with clear examples and exam-style tips.

一元二次方程是IGCSE数学中最重要的考点之一。几乎每份试卷都会出现相关内容，无论是独立题目还是综合应用题。本指南将带你掌握全部必需的解题方法，从因式分解到求根公式，配以清晰的例题和考试技巧。

1. What Is a Quadratic Equation? | 什么是一元二次方程

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The variable x appears with the highest power of 2, and this is what makes the equation quadratic.

一元二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为2，这正是”二次”的含义。

ax² + bx + c = 0 (a ≠ 0)

Examples of quadratic equations include 2x² + 3x − 5 = 0, x² − 9 = 0 and 4x² − 12x + 9 = 0. Notice that in every case the highest exponent of x is exactly 2.

二次方程的例子包括 2x² + 3x − 5 = 0、x² − 9 = 0 和 4x² − 12x + 9 = 0。注意在每种情况下 x 的最高指数都恰好是2。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic expression can be written as a product of two linear factors. If ax² + bx + c = (px + q)(rx + s), then the equation is satisfied when either px + q = 0 or rx + s = 0.

当二次表达式可以写成两个一次因式的乘积时，因式分解是最快捷的方法。如果 ax² + bx + c = (px + q)(rx + s)，那么当 px + q = 0 或 rx + s = 0 时原方程成立。

Example 1 | 例1: Solve x² + 5x + 6 = 0.

Find two numbers that multiply to give 6 and add to give 5. These numbers are 2 and 3, because 2 × 3 = 6

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