📚 Solving Quadratic Equations | 二次方程求解
Quadratic equations appear in almost every Edexcel IGCSE Mathematics examination. From simple factorisation to word problems and graphs, mastering this topic unlocks marks across both Paper 1 and Paper 2. This revision guide explains the key methods step by step, highlights common traps, and provides exam-style practice.
二次方程几乎出现在每一场 Edexcel IGCSE 数学考试中。从简单的因式分解到应用题和图像问题，掌握这个主题能在 Paper 1 和 Paper 2 中带来大量分数。本复习指南会逐步讲解核心方法、指出常见陷阱，并提供考试风格的练习。
1. The Standard Form | 标准形式
Every quadratic equation can be rearranged into the standard form shown below. The coefficients a, b and c are real numbers, and a cannot be zero.
每个二次方程都可以整理为如下所示的标准形式。系数 a、b、c 是实数，且 a 不能为零。
ax² + bx + c = 0, a ≠ 0
ant term. If a = 0, the x² term disappears and the equation becomes linear, not quadratic.
x² 项称为二次项，bx 项称为一次项，c 称为常数项。如果 a = 0，x² 项消失，方程就变成了一次方程，而不是二次方程。
as x² = 6x − 8. You must rearrange it to x² − 6x + 8 = 0 before using any solving method.
考试中你可能会遇到非标准形式的二次方程，例如 x² = 6x − 8。你必须先将其整理为 x² − 6x + 8 = 0，然后才能使用任何求解方法。
2. Solving by Factorisation | 因式分解法
Factorisation means writing the quadratic expression as a product of two brackets. This method works best when the brackets have integer coefficients, and it is usually the fastest approach in the IGCSE exam.
因式分解就是把二次表达式写成两个括号相乘的形式。当括号具有整数系数时，这个方法最为适用，通常也是 IGCSE 考试中最快的求解方法。
Example / 例： Solve x² − 5x + 6 = 0.
(x − 2)(x − 3) = 0
t least one bracket must be zero. Therefore x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.
要使乘积为零，至少有一个括号必须为零。因此 x − 2 = 0 或 x − 3 = 0，解得 x = 2 或 x = 3。
x² − 5x + 6. If the expansion does not match the original equation, look for a sign error.
务必通过展开来检验答案：(x − 2)(x − 3) = x² − 3x − 2x + 6 = x² − 5x + 6。如果展开结果与原方程不一致，请检查符号是否出错。
3. Completing the Square | 配方法
Completing the square is a powerful algebraic technique. For an expression x² + bx, we add and subtract (b/2)² to create a perfect square.
配方法是一种强大的代数技巧。对于表达式 x² + bx，我们加上并减去 (b/2)²，从而构造一个完全平方式。
x² + bx = (x + b/2)² − (b/2)²
Example / 例： Solve x² + 6x + 8 = 0 by completing the square.
(x + 3)² − 9 + 8 = 0
(x + 3)² − 1 = 0 → (x + 3)² = 1
x + 3 = ±1 → x = −2 or x = −4
eting the square also reveals the turning point directly; for this example the vertex of the parabola is (−3, −1).
注意，这些答案与因式分解形式 (x + 2)(x + 4) = 0 一致。配方法还能直接给出抛物线的顶点坐标；本例中顶点为 (−3, −1)。
4. The Quadratic Formula | 求根公式
When factorisation is difficult or impossible, use the quadratic formula. This formula works for every quadratic equation and is given in the Edexcel IGCSE formula booklet.
当因式分解困难或无法进行时，可以使用求根公式。该公式适用于所有二次方程，并在 Edexcel IGCSE 公式册中给出。
x = (−b ± √b² − 4ac) / 2a
Example / 例： Solve 2x² + 3x − 5 = 0. Here a = 2, b = 3, c = −5.
x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2)
x = (−
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