📚 Solving Quadratic Equations and Graphing Quadratic Functions | 解二次方程与二次函数图像
Quadratic equations and functions are a core part of the Edexcel IGCSE Mathematics syllabus. This revision article will guide you through the standard solution methods, the key features of a quadratic graph, and common exam traps.
二次方程与二次函数是Edexcel IGCSE数学考纲的核心内容。本复习文章将带你梳理标准解法、二次图像的关键特征，以及常见的考试陷阱。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2.
二次方程是指形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数是 2。
Examples:
例如：
- 2x² - 3x + 1 = 0 is a quadratic equation. 2x² - 3x + 1 = 0 是一个二次方程。
- x² + 4x = 0 is also quadratic, with c = 0. x² + 4x = 0 也是二次方程，其中 c = 0。
- x³ - 2x = 0 is not quadratic. x³ - 2x = 0 不是二次方程。
2. Solving by Factorisation | 因式分解法
Factorisation is often the quickest method. If you can write ax² + bx + c as (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0.
因式分解通常是最快的方法。如果你能把 ax² + bx + c 写成 (px + q)(rx + s) = 0，那么就有 px + q = 0 或 rx + s = 0。
Worked example: solve x² - 5x + 6 = 0.
示例：解 x² - 5x + 6 = 0。
x² - 5x + 6 = (x - 2)(x - 3) = 0
Therefore x = 2 or x = 3.
因此 x = 2 或 x = 3。
Always expand your brackets to check: (x - 2)(x - 3) = x² - 5x + 6. Time spent checking saves careless errors.
务必展开括号进行验算： (x - 2)(x - 3) = x² - 5x + 6。花一点时间检验，能避免粗心错误。
3. Solving by Completing the Square | 配方法
Completing the square rewrites x² + bx + c as (x + p)² + q. This is useful for solving equations and finding turning points.
配方法将 x² + bx + c 改写成 (x + p)² + q 的形式。这种方法既可用于解方程，也可用于求顶点。
Example: solve x² + 6x + 2 = 0.
例：解 x² + 6x + 2 = 0。
(x + 3)² - 9 + 2 = 0 → (x + 3)² = 7
Take square roots: x + 3 = ±√7, so x = -3 ± √7.
两边开平方： x + 3 = ±√7，所以 x = -3 ± √7。
Remember that the constant in the perfect square is always half of the coefficient of x. For x² + 6x, half of 6 is 3.
请记住：完全平方中的常数项始终是 x 系数的一半。对于 x² + 6x，6 的一半是 3。
4. The Quadratic Formula | 求根公式
For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula.
对于任意二次方程 ax² + bx + c = 0，解可用求根公式给出。
x = (-b ± √b² - 4ac) / (2a)
You must learn this formula for the Edexcel IGCSE exam. Use it when factorisation is impossible or difficult.
你必须为Edexcel IGCSE考试记住这个公式。当无法因式分解或分解较困难时，就使用它。
Example: solve 2x² + 3x - 5 = 0.
例：解 2x² + 3x - 5 = 0。
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