Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. Whether you sit the Core or the Extended paper, you need to know how to factorise, solve, and apply quadratic equations with confidence. This article breaks down every method step by step, with exam-style examples and common pitfalls to avoid.

二次方程是 IGCSE 数学中最常考查的内容之一。无论你参加 Core（核心）还是 Extended（拓展）试卷，都需要熟练掌握二次方程的因式分解、求解与实际应用。本文将逐步拆解每一种方法，并配有考试风格的例题以及需要避免的常见错误。

1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is any equation that can be rearranged into the standard form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. Here x is the unknown variable. If a = 0, the equation becomes linear and is no longer quadratic.

二次方程是指可以整理为标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为实数，且 a ≠ 0。x 为未知数。若 a = 0，方程就变成了一次方程，不再属于二次方程。

Common examples include:

常见的例子包括：

• x² – 7x + 12 = 0

• 2x² + 3x – 5 = 0

• x² – 16 = 0

The solutions of a quadratic equation are also called its roots or zeros. A quadratic equation can have at most two real roots.

二次方程的解也称为它的根或零点。一个二次方程至多有两个实数根。

2. Expanding Brackets: The Foundation | 展开括号：基础技能

Before you can solve quadratics, you must be able to expand double brackets. The rule for expanding (x + p)(x + q) is simple:

在求解二次方程之前，你必须掌握展开双重括号的能力。展开 (x + p)(x + q) 的法则非常简单：

(x + p)(x + q) = x² + (p + q)x + pq

For example, (x + 3)(x – 2) = x² + 3x – 2x – 6 = x² + x – 6. Notice that the coefficient of x comes from adding +3 and -2, while the constant term comes from multiplying them together.

例如，(x + 3)(x – 2) = x² + 3x – 2x – 6 = x² + x – 6。注意 x 的系数来自 +3 与 -2 的相加，而常数项来自它们的相乘。

When the leading coefficient is not 1, expand carefully: (2x + 1)(3x – 4) = 6x² – 8x + 3x – 4 = 6x² – 5x – 4. Practise until this process becomes automatic.

当首项系数不为 1 时，展开要格外仔细：(2x + 1)(3x – 4) = 6x² – 8x + 3x – 4 = 6x² – 5x – 4。反复练习直至这一过程变得非常熟练。

3. Factorising Monic Quadratics | 因式分解首项系数为 1 的二次式

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