📚 Solving Quadratic Equations | 解一元二次方程
Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in both Paper 2 and Paper 4, and a deep understanding of them is essential for success in higher-level mathematics, including Additional Mathematics and A-level.
一元二次方程是 IGCSE 数学大纲中最重要的主题之一。它在 Paper 2 和 Paper 4 中都会出现，深入理解它对于后续更高层次的数学学习（包括附加数学和 A-level）至关重要。
1. What Is a Quadratic Equation? | 什么是一元二次方程
A quadratic equation is an equation that can be written in the general form ax² + bx + c = 0, where a, b and c are constants, x is the unknown variable, and a ≠ 0. The word ‘quadratic’ comes from the Latin word ‘quadratus’, meaning ‘square’, because the highest power of x is x².
一元二次方程是可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，x 是未知数，且 a ≠ 0。”quadratic” 一词源自拉丁语 “quadratus”，意为”平方”，因为 x 的最高次数是 x²。
The condition a ≠ 0 is essential. If a = 0, the equation reduces to a linear equation bx + c = 0, which has only one solution and behaves very differently from a quadratic.
条件 a ≠ 0 至关重要。如果 a = 0，方程就退化为一次方程 bx + c = 0，它只有一个解，性质也与二次方程完全不同。
Examples of quadratic equations include:
一元二次方程的例子包括：
- x² − 5x + 6 = 0
- 2x² + 3x − 1 = 0
- 4x² − 9 = 0
Non-examples include x³ − 2x + 1 = 0, which is a cubic equation, and x + 5 = 0, which is linear.
反例包括 x³ − 2x + 1 = 0，这是一个三次方程；以及 x + 5 = 0，这是一个一次方程。
2. Expanding and Factorising Quadratics | 展开与因式分解二次式
Before solving quadratics, you must be confident with expanding and factorising. Expanding means writing (x + 2)(x + 3) as x² + 5x + 6; factorising is the reverse process, writing x² + 5x + 6 back as (x + 2)(x + 3).
在解二次方程之前，你必须熟练掌握多项式的展开与因式分解。展开意味着把 (x + 2)(x + 3) 写成 x² + 5x + 6；因式分解则是相反的过程，把 x² + 5x + 6 重新写成 (x + 2)(x + 3)。
To factorise x² + bx + c, find two integers whose product is c and whose sum is b. In x² + 5x + 6, the numbers 2 and 3 multiply to give 6 and add to give 5, so the factorised form is (x + 2)(x + 3).
对 x² + bx + c 进行因式分解时，需要找到两个整数，使它们的积等于 c，和等于 b。在 x² + 5x + 6 中，2 和 3 相乘得 6，相加得 5，所以因式分解的结果是 (x + 2)(x + 3)。
You should recognise these special cases from the syllabus:
你应该熟悉大纲中要求的这些特殊公式：
- (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²
(a + b)² = a² + 2ab + b²；(a − b)² = a² − 2ab + b² - a² − b² = (a + b)(a − b), the difference of two squares
a² − b² = (a + b)(a − b)，即平方差公式
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