📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：二次方程完全指南
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graph sketching and real-world modelling. This revision guide covers the essential methods for solving quadratic equations, interpreting the discriminant, and drawing accurate quadratic graphs. It is designed to help you build confidence for both Core and Extended papers.
二次方程在 IGCSE 数学大纲中贯穿始终，从代数变形到函数图像绘制，再到现实情境建模。本复习指南涵盖求解二次方程的基本方法、判别式的意义以及二次函数图像的准确绘制，旨在帮助你在 Core 和 Extended 试卷中建立信心。
1. What is a Quadratic Equation? | 什么是二次方程？
d a ≠ 0. The highest power of the variable x is 2, which gives the equation its characteristic U-shaped graph called a parabola.
二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数为 2，因此其图像具有典型的 U 形，称为抛物线。
Quadratic equations arise naturally when we multiply two linear expressions and set the product equal to zero, or when we model area, projectile motion, and profit functions. Recognising the standard form is the first step in choosing an efficient solution method.
当我们把两个一次因式相乘并令乘积为零，或者对面积、抛体运动和利润函数进行建模时，都会自然地产生二次方程。识别标准形式是选择有效求解方法的第一步。
2. Standard Form and Coefficient Names | 标准式与系数名称
a = 1, the equation is monic; if a ≠ 1, you must factor or use the formula more carefully.
在求解之前，请把每一个二次方程写成标准形式 ax² + bx + c = 0。系数 a 称为首项系数，b 称为一次项系数，c 称为常数项。若 a = 1，该方程为首一方程；若 a ≠ 1，因式分解或使用公式时需要更加仔细。
Example: 3x² − 5x + 2 = 0 has a = 3, b = −5, c = 2.
示例：3x² − 5x + 2 = 0 中，a = 3，b = −5，c = 2。
Always rearrange terms so the right-hand side equals 0.
始终将各项移项，使右边等于 0。
If b = 0 or c = 0, some methods become simpler, but the standard form still applies.
如果 b = 0 或 c = 0，某些方法会变得更简单，但标准形式仍然适用。
3. Solving by Factorising | 因式分解法
d to b. For ax² + bx + c = 0 with a ≠ 1, use splitting the middle term or trial and error.
当二次式可以写成两个一次因式的乘积时，因式分解法是最快的方法。对于 x² + bx + c = 0，寻找两个数，使它们的乘积为 c，和为 b。对于 a ≠ 1 的 ax² + bx + c = 0，可采用中间项拆分法或试误法。
apply the zero-product property: if a product equals zero, at least one factor must equal zero. This gives two linear equations to solve.
一旦因式分解为 (px + q)(rx + s) = 0，应用零乘积性质：若乘积为零，则至少有一个因式为零。由此得到两个一次方程来求解。
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = −2 or x = −3
Factorising also reveals the x-intercepts of the graph y = ax² + bx + c immediately.
因式分解还直接揭示了图像 y = ax² + bx + c 的 x 轴截距。
4. Solving by Completing the Square | 配方法
Completing the square transforms ax² + bx + c into the form a(x + p)² + q. This method is especially useful when the quadratic does not factorise neatly, when finding the turning point, or when deriving the quadratic formula.
配方法将 ax² + bx + c 转化为 a(x + p)² + q 的形式。当二次式不易因式分解、需要求顶点或推导求根公式时，配方法特别有用。
ht, then add (b/2)² to both sides. The left side becomes a perfect square. For a ≠ 1, divide through by a first.
对于 x² + bx + c = 0，先把 c 移到右边，然后两边同时加上 (b/2)²。左边变成完全平方。若 a ≠ 1，先两边除以 a。
x² + 6x + 4 = 0 → x² + 6x = −4 → x² + 6x + 9 = 5 → (x + 3)² = 5 → x = −3 ± √5
re also gives the coordinates of the vertex: for y = a(x + p)² + q, the vertex is (−p, q).
配方法还能给出顶点的坐标：对于 y = a(x + p)² + q，顶点为 (−p, q)。
5. The Quadratic Formula | 求根公式
The formula is derived by completing the square on the general quadratic ax² + bx + c = 0.
求根公式可以求解任何二次方程，包括具有无理根或复数根的方程
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