Mastering Quadratic Equations: Factorisation, Completing the Square and the Quadratic Formula | 掌握二次方程：因式分解、配方法与求根公式

📚 Mastering Quadratic Equations: Factorisation, Completing the Square and the Quadratic Formula | 掌握二次方程：因式分解、配方法与求根公式

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, and real-life modelling. In this revision guide, you will learn how to solve quadratic equations by factorising, completing the square, and using the quadratic formula.

二次方程是 IGCSE 数学中最重要的主题之一。它们出现在代数、图像和现实建模中。在本复习指南中，你将学习如何通过因式分解、配方法和求根公式来解二次方程。

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 variable x is 2, which is why it is called a degree-two polynomial equation.

二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数且 a ≠ 0。变量 x 的最高次数为 2，因此它被称为二次多项式方程。

The name ‘quadratic’ comes from the Latin word quadratus, meaning square. This reminds us that the key feature is the x² term. Examples include x² – 5x + 6 = 0, 2x² + 3x – 2 = 0 and 4x² – 9 = 0.

二次方程的名称来自拉丁语 quadratus，意为正方形。这提醒我们其关键特征是 x² 项。例子包括 x² – 5x + 6 = 0、2x² + 3x – 2 = 0 和 4x² – 9 = 0。

Quadratic equations can have two real roots, one repeated real root, or no real roots. The method you choose depends on the structure of the equation and the instruction in the question.

二次方程可以有两个实根、一个重根或没有实根。你选择的方法取决于方程的结构以及题目中的要求。

2. Standard Form and Rearranging | 标准形式与方程整理

The standard form is ax² + bx + c = 0. It is essential to place all terms on one side and leave zero on the other side before factorising or applying the formula. Many exam mistakes come from trying to factorise when the equation is not equal to zero.

标准形式为 ax² + bx + c = 0。在因式分解或使用公式之前，必须将所有项移到一边，另一边留零。许多考试错误都源于在方程不等于零时尝试因式分解。

For example, rearrange x² + 5x = 6 to x² + 5x – 6 = 0. Here a = 1, b = 5 and c = -6. If the coefficient of x² is negative, you can multiply the whole equation by -1 to make it positive.

例如，将 x² + 5x = 6 整理为 x² + 5x – 6 = 0。这里 a = 1，b = 5，c = -6。如果 x² 的系数为负，你可以将整个方程乘以 -1 使其为正。

Always check that the equation is fully simplified. For instance, 3x² – 6x = 0 becomes 3x(x – 2) = 0, which is easier to solve once factorised. Rearranging correctly is the foundation for all later steps.

务必检查方程是否完全简化。例如，3x² – 6x = 0 变为 3x(x – 2) = 0，因式分解

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