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 tested topics in IGCSE Mathematics. From the Extended paper’s algebraic section to the Core paper’s problem-solving questions, the ability to solve equations of the form ax² + bx + c = 0 is an essential skill. This guide will walk you through every method you need, the logic behind each approach, and the traps that examiners love to set.

二次方程是 IGCSE 数学中考查频率最高的知识点之一。无论是扩展试卷中的代数部分，还是核心试卷中的应用题，掌握 ax² + bx + c = 0 形式的方程求解都是必不可少的能力。本指南将带领你逐一掌握每种解法、其背后的逻辑，以及考官最爱设置的陷阱。

1. The Standard Form and Roots | 标准形式与根

A quadratic equation is any equation that can be rearranged into the standard form:

任何一个可以整理为标准形式的方程都称为二次方程：

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are constants, and a cannot be zero; if a were zero, the equation would become linear. The highest power of x is 2, which is why the graph of a quadratic is a curve called a parabola.

其中 a、b、c 为常数，且 a 不能为零；若 a 为零，方程就变成了一次方程。x 的最高次数为 2，因此二次函数的图像是一条被称为抛物线的曲线。

The solutions to a quadratic equation are called its roots. A quadratic equation always has at most two real roots. Some quadratics have two distinct roots, some have exactly one repeated root, and some have no real roots at all. Finding these roots is the central task of this topic.

二次方程的解称为它的根。一个二次方程至多有两个实数根：有些有两个不同的根，有些恰有一个重根，有些则完全没有实数根。求出这些根正是本专题的核心任务。

2. Solving by Factorisation | 因式分解法

When the quadratic expression can be factorised, this is usually the fastest method. The principle is simple: if two numbers or expressions multiply to give zero, then at least one of them must

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