📚 IGCSE Maths T-5-1014: Solving Quadratic Equations | IGCSE 数学 T-5-1014：解二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, geometry and even probability questions. To solve them confidently, you need to master factorisation, the quadratic formula, and completing the square. This revision guide explains each method step by step and shows common exam-style applications.
二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像、几何甚至概率题中。要自信地解二次方程，你需要掌握因式分解法、公式法和配方法。本复习指南将逐步讲解每种方法，并展示常见的考试题型。
1. What Is a Quadratic Equation? | 什么是二次方程？
condition a ≠ 0 is essential because if a = 0 the equation becomes linear, not quadratic.
一个变量的二次方程次数为 2。它可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。条件 a ≠ 0 很重要，因为如果 a = 0，方程就变成了线性方程，而不是二次方程。
+ 2x − 1 = 0 and x² = 9. Some equations need rearranging before they are in standard form.
例如 x² − 5x + 6 = 0、3x² + 2x − 1 = 0 和 x² = 9。有些方程需要先整理才能变为标准形式。
ax² + bx + c = 0, a ≠ 0
2. Standard Form and Coefficients | 标准形式与系数
to one side so the other side is 0. Identify a, b and c carefully, including their signs.
解二次方程时，首先要把它整理成 ax² + bx + c = 0 的形式。将所有项移到等号一边，使另一边为 0。仔细识别 a、b 和 c，包括它们的符号。
+ 3 = 0, we have a = 2, b = −7 and c = 3. In x² − 4 = 0, we have a = 1, b = 0 and c = −4.
例如，在 2x² − 7x + 3 = 0 中，a = 2、b = −7、c = 3。在 x² − 4 = 0 中，a = 1、b = 0、c = −4。
Getting the signs wrong is a very common mistake, especially when a term is subtracted. Always write the equation in descending powers of x.
符号错误是非常常见的错误，尤其是在减号出现时。务必按照 x 的降幂顺序书写方程。
3. Solving by Factorisation | 因式分解法
+ q)(rx + s) = 0, then either px + q = 0 or rx + s = 0. This is the zero-product property.
当二次表达式可以写成两个一次括号的乘积时，因式分解法适用。如果 (px + q)(rx + s) = 0，那么 px + q = 0 或 rx + s = 0。这就是零乘积性质。
To factorise ax² + bx + c, find two numbers that multiply to give ac and add to give b. Then split the middle term and factorise by grouping.
要对 ax² + bx + c 进行因式分解，需要找到两个数，它们的乘积等于 ac，和等于 b。然后拆分中间项并用分组法分解。
er example is 2x² + 5x − 3 = 0, which becomes (2x − 1)(x + 3) = 0, giving x = ½ or x = −3.
例如，x² − 5x + 6 = 0 分解为 (x − 2)(x − 3) = 0，得到 x = 2 或 x = 3。另一个例子是 2x² + 5x − 3 = 0，它变为 (2x − 1)(x + 3) = 0，得到 x = ½ 或 x = −3。
(x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3
4. Solving by the Quadratic Formula | 公式法
0. It is especially useful when factorisation is difficult or impossible. The formula is:
公式法可以求解任何二次方程 ax² + bx + c = 0 的解。当因式分解困难或不可行时，它特别有用。公式为：
x = (−b ± √b² − 4ac) ÷ 2a
Always write down a, b and c before substituting. Pay special attention to negative signs. If b is negative, −b becomes positive.
代入前务必写下 a、b 和 c。特别注意负号。如果 b 是负数，−b 就变成正数。
For example, solve 2x² + 3x − 2 = 0. Here a = 2,
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