A-Level Edexcel Mathematics: Quadratic Functions – Key Concepts Explained | A-Level Edexcel 数学：二次函数 考点精讲

📚 A-Level Edexcel Mathematics: Quadratic Functions – Key Concepts Explained | A-Level Edexcel 数学：二次函数 考点精讲

Quadratic functions are fundamental to the Edexcel A-Level Mathematics syllabus, appearing in pure topics such as algebraic manipulation, graph sketching, and equation solving, as well as in applied contexts like mechanics and optimisation. Mastering quadratics means understanding their forms, roots, discriminants, and graphical behaviour. This revision guide covers every key concept you need for the exam, from factorising to modelling, with clear bilingual explanations.

二次函数是 Edexcel A-Level 数学教学大纲的基础，出现在代数运算、图像绘制和方程求解等纯数主题中，也出现在力学和优化等应用情境里。掌握二次函数意味着理解其形式、根、判别式和图像行为。本复习指南用清晰的双语解释，覆盖考试所需的每一个关键概念，从因式分解到建模。

1. The Standard Form and Shape of a Quadratic | 二次函数的标准形式和图像形状

A quadratic function is typically written as y = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

二次函数通常写作 y = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。

The graph of a quadratic is a parabola. If a > 0, the parabola opens upwards (∪-shaped) and has a minimum point. If a < 0, it opens downwards (∩-shaped) and has a maximum point.

二次函数的图像是一条抛物线。若 a > 0，抛物线开口向上（∪形），有一个最低点。若 a < 0，开口向下（∩形），有一个最高点。

The coefficient a also controls the width of the parabola; larger |a| makes the graph steeper, while smaller |a| makes it wider.

系数 a 也控制抛物线的宽窄；|a| 越大图像越陡，|a| 越小图像越宽。

2. Factorising Quadratics | 因式分解二次式

Factorising a quadratic expression means writing it as a product of two linear factors: (px + q)(rx + s).

因式分解一个二次式是指将它写成两个一次因式的乘积：(px + q)(rx + s)。

For simple monic quadratics like x² + bx + c, we look for two numbers that multiply to c and add to b. Example: x² + 5x + 6 = (x + 2)(x + 3) because 2×3=6 and 2+3=5.

对于像 x² + bx + c 这样的简单首一二次式，我们寻找两个相乘得 c、相加得 b 的数。例如：x² + 5x + 6 = (x + 2)(x + 3)，因为 2×3=6 且 2+3=5。

When a ≠ 1, we can use the ‘ac method’: multiply a and c, find two numbers that multiply to ac and add to b, rewrite the middle term, and factor by grouping. For instance, 2x² + 7x + 3: ac = 6, numbers 6 and 1. So 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

当 a ≠ 1 时，可使用 ‘ac 法’：将 a 与 c 相乘，找到积为 ac 且和为 b 的两个数，重写中间项，再分组分解。例如 2x² + 7x + 3：ac=6，找到 6 和 1。则 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

Always check your factorisation by expanding.

务必通过展开检查你的因式分解。

3. Solving Quadratic Equations by Factorising | 因式分解法解二次方程

To solve ax² + bx + c = 0 by factorising, first write the quadratic as a product of factors equal to zero, then apply the zero-product property: if AB = 0 then A = 0 or B = 0.

要用因式分解法解 ax² + bx + c = 0，先将二次式写成因式乘积等于零的形式，再应用零积性质：若 AB = 0，则 A = 0 或 B = 0。

Example: x² – 3x – 10 = 0 → (x – 5)(x + 2) = 0 → x = 5 or x = –2.

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