📚 Solving Quadratic Equations: Factorisation, Completing the Square and the Quadratic Formula | IGCSE 数学：二次方程求解——因式分解、配方法与求根公式
Quadratic equations are one of the most important algebra topics in IGCSE Mathematics. You need to be able to solve them by factorisation, by completing the square, and by using the quadratic formula. You also need to understand how the discriminant tells you how many roots an equation has.
二次方程是 IGCSE 数学代数部分最重要的考点之一。你需要掌握因式分解法、配方法和求根公式三种求解方法，还要理解判别式如何判断方程根的数量。
This revision guide explains the key ideas step by step, with worked examples, common mistakes, and exam tips. By the end, you should be able to choose the most efficient method for any quadratic equation you meet in the exam.
本复习指南将逐步讲解核心概念，并配有例题、常见错误和应试技巧。学完后，你将能够在考试中为任意二次方程选择最高效的求解方法。
1. What is a Quadratic Equation? | 什么是二次方程？
ants and a ≠ 0. The highest power of the variable x is 2, which is why it is called a quadratic equation.
二次方程是可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。未知数 x 的最高次数是 2，因此称为二次方程。
quations. Equations such as x³ + 2x² = 0 are not quadratic because the highest power is 3.
例如，2x² − 5x + 3 = 0、x² − 9 = 0 和 4x² + 7x = 0 都是二次方程。x³ + 2x² = 0 不是二次方程，因为最高次数是 3。
x² + bx + c = 0 are the x-coordinates where the graph y = ax² + bx + c crosses the x-axis.
二次方程还与抛物线图像密切相关。方程 ax² + bx + c = 0 的解就是图像 y = ax² + bx + c 与 x 轴交点的横坐标。
2. Standard Form and Coefficients | 标准形式与系数
ecting all terms on one side, usually with the x² term first, then the x term, then the constant term.
在求解二次方程之前，应先将方程整理为标准形式：ax² + bx + c = 0。也就是说，将所有项移到同一边，通常先写 x² 项，再写 x 项，最后写常数项。
The number a is called the leading coefficient, b is the coefficient of x, and c is the constant term. These values are used directly in the quadratic formula and in factorisation, so identifying them correctly is essential.
数字 a 称为首项系数，b 是 x 的系数，c 是常数项。这些数值将直接用于求根公式和因式分解，因此正确识别它们非常重要。
| Equation | a | b | c |
|---|---|---|---|
| 3x² + 5x − 2 = 0 | 3 | 5 | −2 |
| x² − 7 = 0 | 1 | 0 | −7 |
| −4x² + 2x = 0 | −4 | 2 | 0 |
² − 3x + 6 = 0 by subtracting 3x and adding 4 to both sides. Then a = 1, b = −3 and c = 6.
例如，将 3x − 4 = x² + 2 改写为 x² − 3x + 6 = 0，方法是将两边同时减去 3x 并加上 4。此时 a = 1，b = −3，c = 6。
3. Solving by Factorisation | 因式分解法
near brackets. When a = 1, look for two numbers that multiply to give c and add to give b.
当二次式可以写成两个一次括号的乘积时，因式分解通常是最快的方法。当 a =
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