📚 Mastering Quadratic Equations | 掌握二次方程
A quadratic equation is one of the most fundamental topics in IGCSE Mathematics. It appears in nearly every exam paper, often in multiple sections. Understanding how to solve quadratic equations fluently is not just a standalone skill; it is the gateway to graphs, inequalities, and even calculus at higher levels. In this revision guide, we will break down every method you need, with worked examples and exam-style tips.
二次方程是 IGCSE 数学中最基础的主题之一，几乎每份试卷都会考到，而且常常出现在多个板块中。熟练求解二次方程不仅仅是一项独立技能，它更是学习图像、不等式乃至高阶数学内容的敲门砖。在本复习指南中，我们将逐一拆解所有必备解法，并配以例题与考试技巧。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Every quadratic equation can be written in a standard form known as the general form.
二次方程是次数为 2 的多项式方程，即变量的最高次幂为 2。每一个二次方程都可以写成一种称为一般形式的标准形式。
ax² + bx + c = 0, where a ≠ 0
The coefficients a, b and c are real numbers, and a must not be zero. If a = 0, the equation becomes linear rather than quadratic. The term “quadratic” comes from the Latin word “quadratus”, meaning square, because the variable x is raised to the power of 2. A quadratic equation may have two distinct real roots, one repeated real root, or no real roots at all.
系数 a、b、c 均为实数，且 a 不能为零。若 a = 0，方程就变成了一次方程而非二次方程。”Quadratic” 一词源自拉丁语 “quadratus”，意为”正方形”，因为变量 x 被平方。二次方程可能有两个不同的实数根、一个重根，或者没有实数根。
2. Standard Form and Key Terms | 标准形式与关键术语
Before applying any solving method, the equation must first be rearranged into the standard form ax² + bx + c = 0. This means moving all terms to one side of the equals sign. For example, x² = 5x − 6 must be rewritten as x² − 5x + 6 = 0 before any further steps are taken.
在应用任何求解方法之前，方程必须先整理成标准形式 ax² + bx + c = 0。这意味着把所有项移到等号的一侧。例如，x² = 5x − 6 必须改写为 x² − 5x + 6 = 0，然后才能进行后续步骤。
- Quadratic term (二次项): the term ax², which determines the curvature and opening direction of the parabola.
- Linear term (一次项): the term bx, which affects the horizontal position of the vertex.
- Constant term (常数项): the term c, which gives the y-intercept of the graph.
In the equation x² − 5x + 6 = 0, we identify a = 1, b = −5 and c = 6. Always pay close attention to signs when extracting coefficients from a rearranged equation; misreading a negative sign is one of the most frequent errors students make.
在方程 x² − 5x + 6 = 0 中，我们识别出 a = 1，b = −5，c = 6。从整理好的方程中提取系数时，务必高度注意符号；看错负号是学生最常见的错误之一。
3. Solving by Factorisation | 因式分解法
Factorisation is the quickest method when the quadratic has simple integer roots. The idea is to express the quadratic as a product of two linear factors and then use the zero product property: if A × B = 0, then A = 0 or B = 0. This property relies on the fact that the product of two non-zero numbers is never zero.
当二次方程具有简单的整数根时，因式分解是最快的方法。其思想是将二次式写成两个一次因式的乘积，再利用零积性质：若 A × B = 0，则 A = 0 或 B = 0。该性质基于一个事实：两个非零数的乘积不可能为零。
Worked example 1 (例题 1): Solve x² − 5x + 6 = 0.
We look for two numbers that multiply to +6 and add to −5. These numbers are −2 and −3, because (−2) × (−3) = 6 and (−2) + (−3) = −5. Hence the factorised form is:
我们寻找两个数，它们的乘积为 +6，和为 −5。这两个数是 −2 和 −3，因为 (−2) × (−3) = 6 且 (−2) + (−3) = −5。因此分解后的形式为：
(x − 2)(x − 3) = 0
Applying the zero product property gives x − 2 = 0 or x − 3 = 0, so the solutions are x = 2 or x = 3.
应用零积性质得 x − 2 = 0 或 x − 3 = 0，因此解为 x = 2 或 x = 3。
Worked example 2 (例题 2): Solve 2x² + 7x + 3 = 0.
Here the leading coefficient is not 1, so we multiply a and c first: 2 × 3 = 6. We then find two numbers that multiply to 6 and add to 7, which are 1 and 6. Next, split the middle term 7x into x + 6x:
这里首项系数不是 1，因此我们先将 a 与 c 相乘：2 × 3 = 6。然后寻找乘积为 6、和为 7 的两个数，即 1 和 6。接下来将中间项 7x 拆分为 x + 6x：
2x² + x + 6x + 3 = 0
Now factor by grouping. Take x out of the first two terms and 3 out of the last two terms:
现在进行分组分解。从前两项中提出 x，从后两项中提出 3：
x(2x + 1) + 3(2x + 1) = 0
Notice that (2x + 1) is a common factor, so we obtain:
注意 (2x + 1) 是公因式，因此我们得到：
(x + 3)(2x + 1) = 0
Hence x = −3 or x = −½. Always mentally expand your factorised result to check that it reproduces the original quadratic.
因此 x = −3 或 x = −½。务必在心中展开因式分解的结果，以验证是否还原原二次式。
4. Solving by Completing the Square | 配方法
Completing the square is a powerful algebraic technique that rewrites a quadratic in the form a(x + h)² + k. This form
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