📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程完全攻略
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from basic factorising to interpreting real-world projectile motion. A confident grasp of solving methods and graph behaviour will help you score well on both core and extended papers. This guide brings together the essential techniques, common pitfalls, and exam-style thinking you need.
二次方程贯穿 IGCSE 数学课程，从基础的因式分解到解释现实中的抛体运动。熟练掌握各种解法以及图像特征，将帮助你在核心卷和扩展卷中都取得好成绩。本指南汇集了解题所需的核心方法、常见错误和考试思维方式。
1. What is a Quadratic Equation? | 什么是二次方程
A quadratic equation is a polynomial equation in which the highest power of the variable, usually x, is 2. Its general shape involves an x² term, an optional x term, and a constant term.
二次方程是指未知数（通常为 x）的最高次数为 2 的多项式方程。它的一般形式包含 x² 项、可选的 x 项以及一个常数项。
nd c are constants and a ≠ 0. If a were 0, the equation would become linear rather than quadratic, because the x² term would disappear.
在 IGCSE 中，你通常会遇到 ax² + bx + c = 0 这样的方程，其中 a、b、c 为常数且 a ≠ 0。如果 a = 0，方程就变成一次方程，而不是二次方程，因为 x² 项会消失。
adratic is hidden by expanding brackets or by collecting like terms, so always simplify before deciding the type of equation.
例如，2x² - 4x + 1 = 0 是二次方程，但 x³ + x - 2 = 0 不是。二次方程有时会通过展开括号或合并同类项隐藏起来，因此在判断方程类型之前一定要先化简。
2. Standard Form and Identifying Coefficients | 标准形式与系数识别
. This makes it easier to identify the coefficients a, b and c correctly, especially when terms are given in a different order or on different sides of the equals sign.
在求解之前，一定要先把方程整理成标准形式 ax² + bx + c = 0。这样可以更准确地识别系数 a、b 和 c，尤其是当各项顺序被打乱或位于等号两侧时。
rough by -1 gives the equivalent equation 2x² - 3x + 7 = 0, where a = 2, b = -3 and c = 7.
例如，3x - 2x² = 7 应整理为 -2x² + 3x - 7 = 0。各项乘以 -1 后得到等价方程 2x² - 3x + 7 = 0，此时 a = 2，b = -3，c = 7。
= 0. You must include the zero constant when using methods such as the quadratic formula.
另一个常见例子是 4x - x² = 0。它看起来不完整，但仍然是二次方程：a = -1，b = 4，c = 0。在使用求根公式等方法时，必须把零常数项包括进去。
Getting a, b and c the wrong way round is one of the most frequent causes of lost marks, so write them down before doing any calculation.
把 a、b、c 的顺序搞反是最常见的失分原因之一，因此在做任何计算之前，先把它们写下来。
3. Solving by Factorising | 因式分解法
Factorising is often the quickest method when the quadratic has simple integer roots. For a monic quadratic such as x² + bx + c, you look for two numbers that multiply to give c and add to give b.
当二次方程具有较简单的整数根时，因式分解通常是最快的方法。对于 x² + bx + c 这种首项系数为 1 的二次方程，你需要找到两个数，使它们的乘积为 c，和为 b。
+ 3 = 5. Therefore, x² + 5x + 6 = (x + 2)(x + 3), and the solutions are x = -2 or x = -3.
对于 x² + 5x + 6 = 0，两个数是 2 和 3，因为 2 × 3 = 6，2 + 3 = 5。因此，x² + 5x + 6 = (x + 2)(x + 3)，解为 x = -2 或 x = -3。
term: 2x² + 4x + x + 2 = 0. Factor by grouping to get 2x(x + 2) + 1(x + 2) = (2x + 1)(x +
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