Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear in nearly every IGCSE Mathematics paper, whether as a direct solve question, a word problem, or a graph-sketching task. Understanding the three main solving methods — factorisation, the quadratic formula, and completing the square — is essential for securing full marks in this topic. In this revision guide, we walk through each method step by step, examine the discriminant, explore real-life applications, and highlight the traps that students most often fall into during exams.

在 IGCSE 数学考卷中，一元二次方程几乎每卷必考：它可能是直接求解题、文字应用题，也可能是图像作图题。掌握三大解法——因式分解法、求根公式和配方法——是在这一主题上取得满分的关键。在本复习指南中，我们将一步一步拆解每一种解法，深入分析判别式，探索实际应用题，并提醒大家在考试中最常踩中的陷阱。

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

A quadratic equation can always be rearranged into the general form ax² + bx + c = 0, where a, b and c are real constants, x is the unknown, and a ≠ 0. The coefficient a multiplies x², b multiplies x, and c is the constant term. For example, in 3x² − 2x + 5 = 0, we have a = 3, b = −2 and c = 5.

任何一个一元二次方程都可以整理成一般形式 ax² + bx + c = 0，其中 a、b、c 是实数常数，x 是未知数，且 a ≠ 0。系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。例如在 3x² − 2x + 5 = 0 中，a = 3，b = −2，c = 5。

Why must a ≠ 0? If a = 0, the x² term disappears and the equation becomes bx + c = 0, which is a linear equation. It is the presence of the squared term that makes an equation quadratic and, in most cases, gives it two distinct solutions.

为什么 a ≠ 0 呢？如果 a = 0，x² 项就会消失，方程退化为 bx + c = 0，变成一次方程。正是二次项的存在，才使方程具有“二次”的身份，并且在大多数情况下拥有两个不同的解。

2. The Zero Product Property | 零乘积性质

Before learning any solving technique, you must understand the zero product property. It states that if the product of two real numbers or algebraic expressions is zero, then at least one of them must equal zero.

在学习任何解法之前，我们先要理解零乘积性质。它告诉我们：如果两个实数（或两个代数表达式）的乘积等于零，那么其中至少有一个必须等于零。

If A × B = 0, then A = 0 or B = 0

This property turns a quadratic equation into two simple linear equations, which is exactly what we rely on when solving by factorisation. It also explains why a quadratic equation can have two solutions, one solution, or sometimes no real solution at all.

这条性质能把一个二次方程拆成两个简单的一次方程，这正是因式分解法的核心依据。它同时也解释了为什么二次方程可能有两个解、一个解，甚至没有实数解。

3. Solving by Factorisation | 用因式分解法求解

Factorisation is often the fastest method when the coefficients are small and “nice”. A quadratic expression such as x² + bx + c factorises into two linear brackets (x + m)(x + n), provided

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