📚 IGCSE Maths: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
Quadratic equations appear throughout IGCSE Mathematics, from straightforward factorising problems to hidden questions about graphs and real-life modelling. A strong grasp of their structure, solution methods, and graphical interpretation will save time in both core and extended papers.
二次方程贯穿 IGCSE 数学的各个部分，从直接的因式分解题到涉及图像和实际建模的隐含问题。牢固掌握其结构、解法和图像含义，能帮助你在核心和扩展试卷中节省时间。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation in one variable is an equation that can be written in standard form ax² + bx + c = 0 stants and a ≠ 0. The condition a ≠ 0 is essential; if a = 0, the equation becomes linear.
一元二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。a ≠ 0 这一条件至关重要；如果 a = 0，方程就变成了一次方程。
Common examples include 2x² − 3x + 1 = 0 and x² − 9 = 0. Some equations may not look quadratic at first, such as x(x + 4) = 5, but after expanding and rearranging they become quadratic.
常见例子包括 2x² − 3x + 1 = 0 和 x² − 9 = 0。有些方程乍看不是二次方程，例如 x(x + 4) = 5，但展开并整理后就会变成二次方程。
2. Solving by Factorising | 因式分解法
Factorising is usually the fastest method when the quadratic has rational roots. First, rewrite the equation in standard form, then find two numbers whose product is ac and whose sum is b. Split the middle term, factor by grouping, and set each factor equal to zero.
当二次方程有有理根时，因式分解通常是最快的方法。首先将方程写成标准形式，然后找出两个数，使它们的乘积为 ac、和为 b。将中间项拆分，分组分解，再令每个因式等于零。
For example, solve x² + 5x + 6 = 0. The two numbers are 2 and 3, so (x + 2)(x + 3) = 0. Therefore x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.
例如，解 x² + 5x + 6 = 0。这两个数是 2 和 3，所以 (x + 2)(x + 3) = 0。因此 x + 2 = 0 或 x + 3 = 0，得到 x = −2 或 x = −3。
Always expand and simplify first if the equation is not in standard form. For instance, x(x − 2) = 15 becomes x² − 2x − 15 = 0, which factorises as (x − 5)(x + 3) = 0.
如果方程不是标准形式，一定要先展开并化简。例如，x(x − 2) = 15 变为 x² − 2x − 15 = 0，它可因式分解为 (x − 5)(x + 3) = 0。
3. Solving by Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. It is particularly useful for finding the turning point of a graph and for deriving the quadratic formula. Make the coefficient of x² equal to 1, then add and subtract (b/2)².
配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。它对求图像顶点和推导二次公式特别有用。先将 x² 的系数化为 1，然后加上并减去 (b/2)²。
Solve x² + 6x + 5 = 0. This becomes (x + 3)² − 9 + 5 = 0, so (x + 3)² = 4. Taking square roots gives x + 3 = ±2, hence x = −1 or x = −5.
解 x² + 6x + 5 = 0。它变为 (x + 3)² − 9 +
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