📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：掌握二次方程
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation and graph sketching to real-world modelling and problem solving. This revision guide walks through the three main solution methods - factorising, completing the square, and the quadratic formula - and shows how to interpret the discriminant and sketch parabolas with confidence.
二次方程在 IGCSE 数学大纲中无处不在，涉及代数变形、图像绘制以及实际建模和问题解决。本复习指南将讲解三种主要解法——因式分解、配方法和求根公式，并帮助你自信地理解判别式以及绘制抛物线图像。
1. What Is a Quadratic Equation? | 什么是二次方程？
re constants and a ≠ 0. The highest power of the unknown x is 2, which is why it is called 'quadratic' or second-degree.
二次方程是指任何可以写成 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。未知数 x 的最高次数是 2，因此称为“二次”方程。
nging before they match the standard form, such as 2x² = 5x - 3 becoming 2x² - 5x + 3 = 0.
典型例子包括 x² + 5x + 6 = 0、3x² - 12x + 9 = 0，甚至 x² - 4 = 0。有些方程需要先整理才能变成标准形式，例如 2x² = 5x - 3 要先变形为 2x² - 5x + 3 = 0。
+ c. The solutions of the equation are exactly the x-values where the graph crosses the x-axis.
二次方程还与二次函数 y = ax² + bx + c 有密切联系。方程的解正是函数图像与 x 轴交点的横坐标。
2. The Standard Form and Key Terminology | 标准形式与关键术语
fficient of x², b is the coefficient of x, and c is the constant term. The coefficient a must not be zero, otherwise the equation becomes linear rather than quadratic.
二次方程的标准形式是 ax² + bx + c = 0，其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。系数 a 不能为零，否则方程就变成了一次方程而不是二次方程。
- Roots or solutions: values of x that make the equation true.
- Factorising: splitting the quadratic into two linear brackets.
- Discriminant: the expression b² - 4ac, which tells us about the nature of the roots.
- Turning point: the maximum or minimum point on a quadratic graph.
中文对照：根或解是使方程成立的 x 值；因式分解是把二次式拆成两个一次括号；判别式是 b² - 4ac，用来判断根的性质；顶点是二次函数图像上的最高点或最低点。
In IGCSE exam questions, you will often be asked to 'solve' a quadratic equation, which means finding all real values of x that satisfy it. You may also be asked to factorise, complete the square, or sketch the related graph.
在 IGCSE 考试题中，经常会遇到“解”二次方程的要求，也就是找出所有满足方程的实数 x。有时也会要求进行因式分解、配平方或绘制相关图像。
3. Solving by Factorising | 因式分解法
ots. We look for two numbers that multiply to give ac (or c when a = 1) and add to give b.
当二次方程有简单的整数根时，因式分解通常是最快的方法。我们需要找两个数，它们的乘积等于 ac（当 a = 1 时等于 c），它们的和等于 b。
6 and add to 5: 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3) = 0. Therefore x = -2 or x = -3.
例子 1：解 x² + 5x + 6 = 0。我们需要两个数，乘积为 6，和为 5：2 和 3。因此 x² + 5x + 6 = (x + 2)(x + 3) = 0，所以 x = -2 或 x = -3。
is is a difference of two squares, so x² - 9 = (x - 3)(x + 3) = 0, giving x = 3 or x = -3.
例子 2：解 x² - 9 = 0。这是平方差形式，因此 x² - 9 = (x - 3)(x + 3) = 0，得到 x = 3 或 x = -3。
. For example, 2x² + 7x + 3 = 0 factors as (2x + 1)(x + 3) = 0, giving x = -1/2 or x = -3.
当 a 不等于 1 时，可以使用 ac 方法或尝试法。例如 2x² + 7x + 3 = 0 可分解为 (2x + 1)(x + 3) = 0，得到 x = -1/2 或 x = -3。
ys take it out first. For example, 3x² + 6x = 0 becomes 3x(x + 2) = 0, so x = 0 or x = -2.
记得通过展开来检验因式分解是否正确。如果原二次式有公因数，一定要先提取出来。例如 3x² + 6x = 0 变为 3x(x + 2) = 0，所以 x = 0 或 x = -2。
4. Solving by Completing the Square | 配方法
Completing the square is a powerful technique because it works for every quadratic and also helps find the turning point of a graph. The idea is to rewrite ax² + bx + c in the form a(x + p)² + q.
配方法是一种强大的技巧，因为它适用于所有二次方程，也能帮助找出图像的顶点。核心思想是把 ax² + bx + c 改写为 a(x + p)² + q 的形式。
(x + 3)² - 9 + 1 = (x + 3)² - 8. Setting (x + 3)² - 8 = 0 gives (x + 3)² = 8, so x = -3 ± √8.
对于 x² + 6x + 1 = 0，取 6 的一半，即 3，再平方得 9。于是 x² + 6x + 1 = (x + 3)² - 9 + 1 = (x + 3)² - 8。令 (x + 3)² - 8 = 0，得到 (x + 3)² = 8，所以 x = -3 ± √8。
If the coefficient of x²
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