IGCSE Maths Topic 2: Solving Quadratic Equations by Factorising, Completing the Square, and Quadratic Formula | IGCSE 数学主题二：解二次方程——因式分解、配方法与求根公式

📚 IGCSE Maths Topic 2: Solving Quadratic Equations by Factorising, Completing the Square, and Quadratic Formula | IGCSE 数学主题二：解二次方程——因式分解、配方法与求根公式

In IGCSE Mathematics, quadratic equations appear across topics 2, 3 and 6, and they are a core skill for algebra, graphs and coordinate geometry. This article explains how to solve equations of the form ax² + bx + c = 0 by factorising, completing the square, and using the quadratic formula. You will also learn how to interpret the discriminant and avoid common exam errors.

在 IGCSE 数学中，二次方程出现在主题 2、3 和 6 中，是代数、图像和坐标几何的核心技能。本文将讲解如何通过因式分解、配方法和求根公式解形如 ax² + bx + c = 0 的方程。你还将学习如何解释判别式，并避免常见考试错误。

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 standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the unknown x is 2. If a = 0, the equation becomes linear, so a must not be zero for a true quadratic.

二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。未知数 x 的最高次数是 2。如果 a = 0，方程就退化为一次方程，因此真正的二次方程必须满足 a 不为零。

ax² + bx + c = 0, a ≠ 0

ax² + bx + c = 0，a ≠ 0

2. Standard Form and Coefficients | 标准形式与系数

Before solving, always rearrange the equation so that one side is zero and the terms are in descending powers of x. For example, 3x + 2 = x² should be rewritten as x² – 3x – 2 = 0. Here a = 1, b = -3 and c = -2.

在求解之前，一定要把方程整理成一边为零、各项按 x 的降幂排列的形式。例如，3x + 2 = x² 应改写为 x² – 3x – 2 = 0。此时 a = 1，b = -3，c = -2。

The coefficient a is the number multiplied by x²; b is the number multiplied by x; c is the constant term. Identifying a, b and c correctly is essential before using any method.

系数 a 是 x² 前面的数；b 是 x 前面的数；c 是常数项。在使用任何方法之前，正确识别 a、b、c 至关重要。

3. Solving by Factorising: Basic Idea | 因式分解法：基本思想

If a quadratic expression can be factorised into two linear brackets, say (px + q)(rx + s) = 0, then the product is zero. The zero product property states that if A × B = 0, then A = 0 or B = 0. So we set each bracket equal to zero and solve two simple linear equations.

如果二次式可以分解为两个一次因式的乘积，例如 (px + q)(rx + s) = 0，那么乘积为零。零乘积性质指出，如果 A × B = 0，则 A = 0 或 B = 0。因此我们让每个括号分别等于零，再解两个简单的一次方程。

(x + 2)(x – 5) = 0 ⇒ x = -2 or x = 5

(x + 2)(x – 5) = 0 ⇒ x = -2 或 x = 5

4. Factorising When a = 1 | 首项系数为 1 的因式分解

When a = 1, look for two numbers that multiply

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