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 central to the IGCSE Mathematics course. A solid understanding of their structure and solution methods is essential for success in algebra, graph sketching and many real-world applications. This revision guide introduces the standard form of a quadratic, shows three main methods of solving it, and explains how the discriminant helps you predict the number of roots without solving the equation.

二次方程是 IGCSE 数学课程的核心内容。牢固掌握二次方程的结构与求解方法，对于代数、函数图像以及许多实际应用中的成功至关重要。本复习指南将介绍二次方程的标准形式，展示三种主要求解方法，并说明如何利用判别式在不求解的情况下判断根的数量。

1. Recognising a Quadratic Equation | 识别二次方程

A quadratic equation is an equation in which the highest power of the variable is 2. It can be written in the general form ax² + bx + c = 0, where a, b and c are constants and a is not zero. If a were zero, the equation would become linear rather than quadratic.

二次方程是指变量的最高次数为 2 的方程。它可以写成一般形式 ax² + bx + c = 0，其中 a、b、c 为常数，且 a 不等于零。如果 a 等于零，方程就变成一次方程而不是二次方程。

• Example: 3x² + 2x – 5 = 0 is quadratic.

  示例：3x² + 2x – 5 = 0 是二次方程。

• Example: 5x – 7 = 0 is not quadratic, because the highest power of x is 1.

  示例：5x – 7 = 0 不是二次方程，因为 x 的最高次数是 1。

2. The Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0

Before applying any solving method, rearrange the equation so that all terms are on one side and the other side is zero. The terms should be written in descending powers of x: ax² + bx + c = 0. This standard form makes factorisation, completing the square and the quadratic formula easier to apply.

在应用任何求解方法之前，应先将方程整理为所有项都在等号一侧、另一侧为零。各项应按 x 的降幂排列：ax² + bx + c = 0。这个标准形式会使因式分解、配方法和求根公式的应用更加简便。

For example, x² + 3x = 10 must be rearranged to x² + 3x – 10 = 0. Only then can you reliably factorise or identify the values of a, b and c.

例如，x² + 3x = 10 必须先整理为 x² + 3x – 10 = 0。只有这样，你才能准确地进行因式分解或确定 a、b、c 的值。

3. Solving by Factorisation | 因式分解法

If a quadratic expression factorises into two linear brackets, you can use the zero product property. This states that if (px + q)(rx + s) = 0, then at least one factor must equal zero: px + q = 0 or rx + s = 0. Solving each linear equation gives the roots of the original quadratic.

如果一个二次式可以因式分解为两个一次括号，就可以使用零乘积性质。该性质

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