📚 IGCSE Mathematics: Quadratic Equations from Teacher's Guide 400 | IGCSE 数学：教师用书400之二次方程
Quadratic equations form one of the most heavily examined topics in IGCSE Mathematics. They connect algebraic manipulation, coordinate geometry, and real-world modelling, so a strong grasp of quadratics gives you an advantage across multiple question types.
二次方程是 IGCSE 数学中考查频率最高的主题之一。它将代数运算、坐标几何和现实建模联系在一起，因此扎实掌握二次方程能让你在多种题型中占据优势。
1. What Is a Quadratic Equation? | 什么是二次方程？
be written in the standard form ax² + bx + c = 0 where a, b and c are constants and a ≠ 0.
二次方程是一个次数为 2 的多项式方程。未知变量的最高次数是 2，通常可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。
ratic but linear, so the condition a ≠ 0 is essential. In the IGCSE examination, you may be asked to identify whether an equation is quadratic by rearranging terms.
如果 a = 0，方程就不再是二次方程而是线性方程，因此 a ≠ 0 这一条件非常重要。在 IGCSE 考试中，你可能会遇到通过移项来判断方程是否为二次方程的题目。
2. Standard Form and Coefficients | 标准形式与系数
u solve it. The numbers a, b and c are called coefficients: a is the coefficient of x², b is the coefficient of x, and c is the constant term.
每个二次方程在求解前都应先整理成 ax² + bx + c = 0。数字 a、b 和 c 被称为系数：a 是 x² 的系数，b 是 x 的系数，c 是常数项。
en as 2x - x² = 4, first rearrange it to -x² + 2x - 4 = 0 or equivalently x² - 2x + 4 = 0.
例如，方程 3x² - 5x + 2 = 0 中 a = 3、b = -5、c = 2。如果给出的方程是 2x - x² = 4，应先将它移项为 -x² + 2x - 4 = 0，或等价地写成 x² - 2x + 4 = 0。
- English: Always check the sign of each coefficient after rearranging.
- 中文：移项后务必检查每个系数的正负号。
- English: Keep a positive if possible; multiply the whole equation by -1 if needed.
- 中文：如果可能，让 a 为正数；必要时可以将整个方程乘以 -1。
3. Solving by Factorising | 因式分解法求解
Factorising is often the fastest method when the quadratic expression has simple integer factors. The idea is to write ax² + bx + c as a product of two binomials and then set each factor equal to zero.
当二次式的整数因式比较简单时，因式分解通常是最快的方法。思路是将 ax² + bx + c 写成两个一次二项式的乘积，然后令每个因式等于零。
or x = 4. Always remember that if (x - p)(x - q) = 0, then either x - p = 0 or x - q = 0.
例如，x² - 7x + 12 = 0 可因式分解为 (x - 3)(x - 4) = 0，从而得到 x = 3 或 x = 4。始终记住，如果 (x - p)(x - q) = 0，那么 x - p = 0 或 x - q = 0。
ce with different forms such as 2x² + 5x - 3 = 0, which factorises to (2x - 1)(x + 3) = 0.
如果系数 a 不等于 1，你可能需要使用拆中项法或试错法。多练习不同形式，例如 2x² + 5x - 3 = 0 可因式分解为 (2x - 1)(x + 3) = 0。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x - h)² + k. This is especially useful when the quadratic does not factorise neatly or when you need to find the vertex of a parabola.
配方法将二次式改写为 a(x - h)² + k 的形式。当二次式不容易因式
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