📚 Mastering Quadratic Equations for IGCSE Maths | IGCSE 数学二次方程完全掌握
A quadratic equation is one of the most important algebraic topics in IGCSE Mathematics. It appears in Paper 2 and Paper 4, both as direct solving questions and as part of graphs, inequalities and real-life modelling. This revision guide covers every method you need: factorising, completing the square, the quadratic formula, the discriminant and graph interpretation.
二次方程是 IGCSE 数学中最重要的代数主题之一。它在 Paper 2 和 Paper 4 中都会出现，既可能直接求解，也会与图像、不等式和实际建模题结合。本复习指南涵盖你需要掌握的每一种方法：因式分解法、配方法、求根公式、判别式以及图像解读。
1. What Is a Quadratic Equation? | 什么是二次方程？
nstants and a ≠ 0. If a were zero, the equation would become linear rather than quadratic.
二次方程是指次数为二的方程。其标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。如果 a 等于零，方程就会退化为一次方程，而不是二次方程。
nd x² - 9 = 0. Some equations may need rearrangement before you can see the standard form.
典型例子包括 x² - 5x + 6 = 0、2x² + 3x - 2 = 0 和 x² - 9 = 0。有些方程需要先移项整理，才能看出标准形式。
2. The Roles of a, b and c | 系数 a、b、c 的作用
ry and position, while the constant c gives the y-intercept of the graph y = ax² + bx + c.
首项系数 a 决定抛物线的形状和开口方向。系数 b 影响对称轴和图像位置，而常数项 c 表示图像 y = ax² + bx + c 在 y 轴上的截距。
x + 7 = 0, we have a = 2, b = -4 and c = 7. Always include the sign with each coefficient.
在确定 a、b、c 之前，一定要先把方程写成标准形式。例如，在 2x² - 4x + 7 = 0 中，a = 2，b = -4，c = 7。每个系数的符号都要一起保留。
3. Solving by Factorising | 因式分解法
Factorising is often the fastest method when the quadratic has simple rational roots. First set one side to zero, then factorise the other side into two brackets, and finally set each bracket equal to zero.
当二次方程具有简单的有理根时，因式分解通常是最快的方法。首先将方程一边化为零，然后把另一边分解为两个括号的乘积，最后令每个括号分别等于零。
add to -7, so (x - 3)(x - 4) = 0. Therefore x - 3 = 0 or x - 4 = 0, giving x = 3 or x = 4.
例如，解 x² - 7x + 12 = 0。我们需要两个数，乘积为 12，和为 -7，因此得到 (x - 3)(x - 4) = 0。于是 x - 3 = 0 或 x - 4 = 0，解得 x = 3 或 x = 4。
nd add to 7, namely 6 and 1, so 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
当 a 不等于 1 时，可以使用 ac 法或分解中项。对于 2x² + 7x + 3，我们需要两个数，乘积为 2 × 3 = 6，和为 7，也就是 6 和 1，因此 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
4. Solving by Completing the Square | 配方法
Completing the square is useful when factorising is difficult or when you need to find the turning point. The goal is to rewrite x² + bx + c in the form (x + p)² + q.
配方法在因式分解困难或需要求抛物线顶点时非常有用。其目标是把 x² + bx + c 改写成 (x + p)² + q 的形式。
To complete the square for x
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