Solving Quadratic Equations: Factorising, Formula and Completing the Square | 解二次方程：因式分解、公式法与配方法

📚 Solving Quadratic Equations: Factorising, Formula and Completing the Square | 解二次方程：因式分解、公式法与配方法

Quadratic equations appear in nearly every IGCSE Mathematics paper, usually as a short question, a graph question, or part of a word problem. In this revision guide, we will cover the three standard methods for solving a quadratic equation: factorising, the quadratic formula, and completing the square. We will also explain the discriminant and highlight the most common exam traps.

二次方程几乎出现在每一份 IGCSE 数学试卷中，常见题型包括简答题、图象题或应用题的一部分。在本复习指南中，我们将系统讲解解二次方程的三种标准方法：因式分解法、公式法和配方法，并解释判别式的作用，指出最常考也最容易出错的地方。

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

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of x is 2, which is why the equation is called “quadratic”.

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。未知数 x 的最高次数是 2，因此被称为“二次”方程。

The graph of a quadratic is a curve called a parabola. When a is positive, the parabola opens upwards; when a is negative, it opens downwards. The solutions of the equation are the x-coordinates of the points where the parabola crosses the x-axis.

二次函数的图象是一条称为“抛物线”的曲线。当 a 为正数时，抛物线开口向上；当 a 为负数时，开口向下。方程的解就是抛物线与 x 轴交点的横坐标。

Not every quadratic equation has two distinct solutions. Some have one repeated solution, and some have no real solutions at all. We will see how to decide this using the discriminant later in this guide.

并非每个二次方程都有两个不同的解。有些方程只有一个重根，有些则完全没有实数解。我们稍后会用判别式来判断根的情况。

2. Method 1: Factorising When a = 1 | 方法一：二次项系数为 1 的因式分解

When a = 1, we look for two numbers that multiply to give c and add to give b. If those numbers are p and q, then x² + bx + c = (x + p)(x + q). This is the fastest method when the factor pair is easy to spot.

当 a = 1 时，我们寻找两个数 p 和 q，使它们的乘积等于 c、和等于 b。此时 x² + bx + c = (x + p)(x + q)。当数字比较容易找时，这是最快的方法。

Example: Solve x² + 5x + 6 = 0. We need two numbers whose product is 6 and whose sum is 5. Those numbers are 2 and 3. Hence x² + 5x + 6 = (x + 2)(x + 3) = 0.

例：解 x² + 5x + 6 = 0。我们需要两个数，乘积为 6、和为 5。这两个数是 2 和 3，因此 x² + 5x + 6 = (x + 2)(x + 3) = 0。

(x + 2)(x + 3) = 0, so x + 2 = 0 or x + 3 = 0. Hence x = -2 or x = -3.

(x + 2)(x + 3) = 0，所以 x + 2 = 0 或 x + 3 = 0，因此 x = -2 或 x = -3。

The key rule is the zero-product property: if the product of two factors is zero, then at least one of the factors must be zero. This is why factorising works as a solving method.

关键规则是“零乘积性质”：若两个因式的乘积为零，则至少有一个因式为零。这正是因式分解可以求解方程的原理。

3. Factorising Special Cases | 因式分解的特殊情形

Before attempting to factorise any quadratic, always check whether there is a common factor. For example, 3x² – 6x – 9 = 3(x² – 2x – 3) = 3(x – 3)(x + 1). Removing a common factor first makes the rest much easier.

分解任何二次式之前，务必先检查是否有公因式。例如，3x² – 6x – 9 = 3(x² – 2x – 3) = 3(x – 3)(x + 1)。先提取公因式会让后续分解轻松得多。

Another important pattern is the difference of two squares: x² – a² = (x – a)(x + a). For example, x² – 25 = (x – 5)(x + 5), and 4x² – 9 = (2x – 3)(2x + 3).

另一个重要公式是平方差公式：x² – a² = (x – a)(x + a)。例如，x² – 25 = (x – 5)(x + 5)，4x² – 9 = (2x – 3)(2x + 3)。

When the coefficient a is not 1, such as 2x² + 7x + 3, we can use the “multiply and split” method. First multiply a and c: 2 × 3 = 6. Then find two numbers whose product is 6 and whose sum is 7; those numbers are 1 and 6.

当二次项系数 a 不为 1 时，例如 2x² + 7x + 3，可以采用“相乘拆分”的方法。先将 a 与 c 相乘：2 × 3 = 6，然后找出乘积为 6、和为 7 的两个数，即 1 和 6。

Rewrite the middle term and group: 2x² + x + 6x + 3 = x(

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