Formula Handbook for IB & Edexcel Mathematics | IB 与 Edexcel 数学公式汇总手册

📚 Formula Handbook for IB & Edexcel Mathematics | IB 与 Edexcel 数学公式汇总手册

This comprehensive formula handbook covers essential formulas for both IB Mathematics (Analysis & Approaches and Applications & Interpretation) and Edexcel A Level Mathematics (Pure, Statistics, and Mechanics). Use it as a quick reference guide for revision and examination preparation.

本公式手册涵盖了 IB 数学（分析与方法、应用与解释）以及 Edexcel A Level 数学（纯数、统计与力学）的核心公式，可作为复习和备考的速查手册。

1. Algebra and Exponents | 代数与指数定律

Fundamental rules of indices for simplifying algebraic expressions:

简化代数表达式的基本指数法则：

a^m × aⁿ = a^m+n

a^m ÷ aⁿ = a^m−n

(a^m)ⁿ = a^mn

a⁰ = 1 (a ≠ 0)

a⁻ⁿ = 1 / aⁿ

a^1/n = √[n]{a}

These laws are valid for all real numbers a, b and integers m, n. The binomial expansion and Pascal’s triangle are also crucial tools in algebraic manipulation.

这些定律适用于所有实数 a、b 和整数 m、n。二项式展开和帕斯卡三角形也是代数处理中的关键工具。

For Edexcel and IB HL, the binomial theorem for a rational index: (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … , provided |x|<1.

对于 Edexcel 和 IB HL，有理数指数的二项式定理：(1+x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …，其中 |x|<1。

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

The standard quadratic equation is ax² + bx + c = 0, with a ≠ 0.

标准二次方程为 ax² + bx + c = 0，且 a ≠ 0。

x = [ −b ± √(b² − 4ac) ] / (2a)

The discriminant Δ = b² − 4ac determines the nature of the roots:

判别式 Δ = b² − 4ac 决定了根的性质：

• Δ > 0: two distinct real roots | 两个不相等的实根
• Δ = 0: one repeated real root | 一个重实根
• Δ < 0: no real roots (complex conjugates) | 无实根（共轭复数）

Sum of roots α + β = −b/a ; product αβ = c/a. These relations are used in both IB and Edexcel.

根的和 α + β = −b/a；根的积 αβ = c/a。这些关系在 IB 和 Edexcel 考试中均常用。

Completing the square transforms ax² + bx + c into a(x + b/(2a))² + (c − b²/(4a)).

配方法将 ax² + bx + c 变形为 a(x + b/(2a))² + (c − b²/(4a))。

3. Sequences and Series | 序列与级数

Arithmetic sequence: nth term u_n = a + (n−1)d. Sum of first n terms S_n = n/2 [2a + (n−1)d] or S_n = n/2 (a + l), where l is the last term.

等差数列：第 n 项 u_n = a + (n−1)d。前 n 项和 S_n = n/2 [2a + (n−1)d] 或 S_n = n/2 (a + l)，l 为末项。

Geometric sequence: u_n = arⁿ⁻¹. Sum of n terms S_n = a(1 − rⁿ)/(1 − r) for r ≠ 1. Sum to infinity S_∞ = a/(1 − r) provided |r| < 1.

等比数列：u_n = arⁿ⁻¹。前 n 项和 S_n = a(1 − rⁿ)/(1 − r)，r ≠ 1。无穷级数和 S_∞ = a/(1 − r)，要求 |r| < 1。

Standard sigma notation results often required in Edexcel and IB: Σ_k=1ⁿ k = n(n+1)/2, Σ_k=1ⁿ k² = n(n+1)(2n+1)/6, Σ_k=1ⁿ k³ = [n(n+1)/2]².

标准求和符号的结果常在 Edexcel 与 IB 中出现：Σ_k=1ⁿ k = n(n+1)/2，Σ_k=1ⁿ k² = n(n+1)(2n+1)/6，Σ_k=1ⁿ k³ = [n(n+1)/2]²。

4. Trigonometry Identities and Equations | 三角恒等式与方程

Core Pythagorean identity

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