Edexcel A-Level Pure Maths: Core Techniques and Worked Examples | 爱德思 A-Level 纯数学：核心方法与例题

📚 Edexcel A-Level Pure Maths: Core Techniques and Worked Examples | 爱德思 A-Level 纯数学：核心方法与例题

This revision article brings together the core pure-mathematics techniques that regularly appear in Edexcel A-Level Mathematics Paper 1 and Paper 2. It is designed around the mixed practice page ‘pdfjoiner (4)-076’ and covers algebra, functions, trigonometry, calculus, sequences, coordinate geometry and vectors.

本篇复习文章汇集了爱德思 A-Level 数学 Paper 1 和 Paper 2 中经常出现的纯数学核心方法。文章围绕综合练习页 ‘pdfjoiner (4)-076’ 设计，涵盖代数、函数、三角、微积分、数列、坐标几何与向量等内容。

1. Algebraic Simplification and Surds | 代数化简与根式

When simplifying surds, always look for square factors and rationalise denominators where possible.

化简根式时，应始终寻找平方因子，并尽可能有理化分母。

√a × √b = √(ab)

For example, √48 can be written as √(16 × 3) = 4√3.

例如，√48 可以写成 √(16 × 3) = 4√3。

To rationalise a denominator such as 1/(√3 + √2), multiply the numerator and denominator by the conjugate √3 – √2.

要化简 1/(√3 + √2) 这样的分母，将分子和分母同时乘以共轭根式 √3 – √2。

1/(√3 + √2) × (√3 – √2)/(√3 – √2) = (√3 – √2)/(3 – 2) = √3 – √2

In algebraic fractions, factorise both numerator and denominator first, then cancel common factors.

在代数分式中，应先对分子和分母因式分解，然后约去公因式。

2. Quadratic Functions and the Discriminant | 二次函数与判别式

The discriminant of ax² + bx + c = 0 is given by Δ = b² – 4ac.

二次方程 ax² + bx + c = 0 的判别式为 Δ = b² – 4ac。

It tells us how many real roots the equation has without solving it fully.

它使我们无需完整求解方程即可判断实根的个数。

Δ > 0: two distinct real roots; Δ = 0: one repeated real root; Δ < 0: no real roots.

Δ > 0：两个不同实根；Δ = 0：一个重根；Δ < 0：无实根。

For example, x² – 5x + 3 = 0 has Δ = (-5)² – 4(1)(3) = 25 – 12 = 13, so it has two distinct real roots.

例如，x² – 5x + 3 = 0 的判别式 Δ = (-5)² – 4(1)(3) = 25 – 12 = 13，因此它有两个不同实根。

Completing the square transforms a quadratic into the form a(x + p)² + q, which reveals the vertex (-p, q).

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