📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学中的二次方程
the main solution methods will help you answer exam questions confidently and accurately.
二次方程是 IGCSE 数学课程中最重要的代数主题之一。它在核心和扩展试卷中都会出现，并且经常与图像、代数变形和实际问题联系在一起。扎实掌握标准形式 ax² + bx + c = 0 以及主要求解方法，能帮助你自信、准确地解答考试题目。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation in which the highest power of the variable is 2. It usually contains one variable, often written as x, and can be expressed in the general form shown below.
二次方程是一个最高次数为 2 的多项式方程。它通常只含有一个变量，常用 x 表示，并且可以写成下面的一般形式。
ax² + bx + c = 0, where a ≠ 0
The condition a ≠ 0 is essential because if a were 0, the equation would become linear rather than quadratic. The coefficients a, b and c can be positive or negative numbers, fractions, decimals or even algebraic expressions in some harder questions.
条件 a ≠ 0 非常重要，因为如果 a 为 0，方程就变成一次方程，而不是二次方程。系数 a、b 和 c 可以是正数、负数、分数、小数，在一些较难的题目中甚至可以是代数式。
arrange it into standard form before identifying a, b and c or choosing a solution method.
在 IGCSE 题目中，二次方程一开始可能以变形形式出现，例如 3x² = 4x - 2。你必须先把它整理成标准形式，然后再识别 a、b 和 c 或选择求解方法。
2. Standard Form and Coefficients | 标准形式与系数
Writing the equation in standard form means placing all terms on one side of the equals sign and zero on the other side. This step makes the values of a, b and c clear and reduces sign errors later in the working.
把方程写成标准形式，就是把所有项都移到等号一边，另一边为 0。这一步能让 a、b 和 c 的值变得清晰，并减少后续解题中的符号错误。
e it as 3x² - 4x - 5 = 0. In this form, a = 3, b = -4 and c = -5, with the signs included.
例如，要解 3x² - 4x = 5，先把它改写为 3x² - 4x - 5 = 0。在这个形式中，a = 3，b = -4，c = -5，符号必须包含在内。
or instance, (x + 1)² = 4 becomes x² + 2x + 1 = 4 and then x² + 2x - 3 = 0 before solving.
如果方程含有括号或分数，要先展开或化简。例如 (x + 1)² = 4 先变成 x² + 2x + 1 = 4，再变成 x² + 2x - 3 = 0，然后才能求解。
Many mistakes in exams come from rushing this stage, so take a moment to check that every sign and term has been moved correctly before proceeding.
考试中的许多错误都来自这一阶段过于匆忙，所以在继续求解之前，花一点时间检查每一项和符号是否都移动正确。
3. Solving by Factorisation | 因式分解法求解
Factorisation is usually the fastest method when the quadratic can be written as the product of two linear factors. The aim is to express ax² + bx + c in the form (px + q)(rx + s), where p, q, r and s are integers in most IGCSE questions.
当二次式可以写成两个一次因式的乘积时，因式分解法通常是最快的方法。目标是把 ax² + bx + c 表示成 (px + q)(rx + s) 的形式，在多数 IGCSE 题目中 p、q、r、s 都是整数。
and -3 work because -2 × -3 = 6 and -2 + -3 = -5, so the factorisation is (x - 2)(x - 3).
对于首项系数为 1 的简单二次式，例如 x² - 5x + 6，要找出两个数，使它们的乘积为 c = 6，和为 b = -5。数字 -2 和 -3 符合要求，因为 -2 × -3 = 6，-2 + -3 = -5，所以因式分解为 (x - 2)(x - 3)。
and 6. Split the middle term: 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3).
对于首项系数不为 1 的二次式，例如 2x² + 7x + 3，你可以使用 ac 法。把 a 和 c 相乘得到 6，然后找出两个数，它们的乘积为 6，和为 7：1 和 6。拆开中间项：2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)。
Always expand your factors mentally to check that they give the original quadratic. This quick verification can prevent lost marks from a careless factor pair.
一定要在心里展开你的因式，检查它们是否能还原原来的二次式。这个快速验证可以防止因粗心选错因数而丢分。
4. Zero Product Property | 零乘积性质
The zero product property states that if the product of two expressions is zero, then at least one of the expressions must be equal to zero. This principle is the reason why factorisation works for solving quadratic equations.
零乘积性质指出，如果两个表达式的乘积为零，那么至少有一个表达式必须等于零。这个原理就是因式分解法能解二次方程的原因。
example, if (x - 2)(x - 3) = 0, then either x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.
一旦二次式被因式分解，就让每个因式分别等于零。例如，如果 (x - 2)(x - 3) = 0，那么 x - 2 =
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