Maths Formula Booklet | 数学公式手册

📚 Maths Formula Booklet | 数学公式手册

This article provides a concise, exam-focused reference for the essential formulas used in A-level and IGCSE Mathematics. Keep this booklet by your side when doing past papers, and use it to check definitions, identities, and standard results before you enter the exam room. It is not a replacement for understanding, but a revision aid to make recall faster and more accurate.

本文提供一份简洁、紧扣考点的核心数学公式参考，适用于 A-level 和 IGCSE 数学。做历年真题时请把它放在手边，用来核对定义、恒等式和标准结论，帮助你在考前快速回忆。它不是理解的替代品，而是让记忆更快、更准的复习工具。

1. Algebraic Identities and Surds | 代数恒等式与根式

Start with the three binomial identities: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b². These are used constantly in expansion, factorisation, and completing the square.

先掌握三个二项式恒等式：(a + b)² = a² + 2ab + b²、(a − b)² = a² − 2ab + b²，以及 (a + b)(a − b) = a² − b²。它们在展开、因式分解和配方法中会反复使用。

Surds follow √(ab) = √a × √b and √(a/b) = √a/√b. To rationalise a denominator such as 1/√a, multiply numerator and denominator by √a to get √a/a. Index laws include a^maⁿ = a^m+n, (a^m)ⁿ = a^mn, a⁰ = 1, and a⁻ⁿ = 1/aⁿ for a ≠ 0.

根式运算遵循 √(ab) = √a × √b 和 √(a/b) = √a/√b。要把 1/√a 这样的分母有理化，可将分子分母同乘 √a，得到 √a/a。指数律包括 a^maⁿ = a^m+n、(a^m)ⁿ = a^mn、a⁰ = 1，以及 a⁻ⁿ = 1/aⁿ（a ≠ 0）。

2. Quadratic Equations and Inequalities | 二次方程与不等式

For ax² + bx + c = 0, the roots are given by x = [−b ± √(b² − 4ac)] / (2a). The discriminant Δ = b² − 4ac determines the nature: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.

对于 ax² + bx + c = 0，其根为 x = [−b ± √(b² − 4ac)] / (2a)。判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不同实根，Δ = 0 有一个重根，Δ < 0 没有实根。

Complete the square by rewriting ax² + bx + c as a(x + b/2a)² + (c − b²/4a). This form shows the vertex of the parabola: (−b/2a, c − b²/4a). For quadratic inequalities such as ax² + bx + c > 0, sketch the graph and test intervals; the sign changes only at real roots.

Published by TutorHao | Mathematics Revision Series | aleveler.com

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