Mastering Quadratic Equations for IGCSE Mathematics | 掌握IGCSE数学中的二次方程

📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握IGCSE数学中的二次方程

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graph sketching and real-world modelling. A strong command of quadratics helps you solve equations, interpret graphs, and tackle extended questions confidently. This article covers recognition, factorisation, completing the square, the quadratic formula, the discriminant, graphs, applications, and common exam traps.

二次方程贯穿IGCSE数学大纲，从代数变形到图像绘制再到实际问题建模。牢固掌握二次方程能帮助你解方程、解读图像并自信应对扩展题。本文涵盖识别二次方程、因式分解法、配方法、求根公式、判别式、图像、应用以及常见考试陷阱。

1. Recognising Quadratic Equations | 识别二次方程

A quadratic equation in one variable is an equation of degree 2, usually written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is essential: if a = 0, the equation becomes linear, not quadratic.

一元二次方程是次数为 2 的方程，通常写作 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。条件 a ≠ 0 很关键：如果 a = 0，方程就变成了一次方程，而不是二次方程。

For example, x² – 5x + 6 = 0 is quadratic, while 2x + 3 = 0 is linear. IGCSE questions often ask you to solve quadratic equations or to identify the coefficients a, b and c before using a formula.

例如，x² – 5x + 6 = 0 是二次方程，而 2x + 3 = 0 是一次方程。IGCSE 题目常要求解二次方程，或在使用公式前先识别系数 a、b、c。

You must also be able to spot quadratics that are hidden by rearrangement, such as x² + 3 = 5x + 1. Once terms are collected on one side, the equation becomes x² – 5x + 2 = 0.

你还必须能够识别经过移项后隐藏的二次方程，例如 x² + 3 = 5x + 1。把所有项移到一边后，方程变为 x² – 5x + 2 = 0。

2. Writing Quadratics in Standard Form | 将二次式写成标准形式

Before solving, rewrite the quadratic as ax² + bx + c = 0, with all terms on one side and zero on the other. This step makes factorisation, completing the square, and formula substitution much easier.

求解前，要先把二次方程写成 ax² + bx + c = 0 的标准形式，使所有项在一边，另一边为零。这一步会让因式分解、配方法和代入公式变得容易得多。

For example, 4x² – 8 = 2x² + 7x can be rearranged to 2x² – 7x – 8 = 0 by subtracting 2x² and 7x from both sides. Always collect like terms carefully and watch the signs.

例如，4x² – 8 = 2x² + 7x 可以通过两边减去 2x² 和 7x 整理为 2x² – 7x – 8 = 0。务必仔细合并同类项并注意符号。

In standard form, a is the coefficient of x², b is the coefficient of x, and c is the constant term. Identifying these values correctly is especially important for the quadratic formula and the discriminant.

在标准形式中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。正确识别这些值对于求根公式和判别式尤为重要。

3. Solving by Factorisation | 因式分解法求解

Factorisation is usually the fastest method when the quadratic has integer roots. To solve ax² + bx + c = 0, factor

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