📚 Essential Formula Guide for Edexcel A-Level Maths | Edexcel A-Level 数学必备公式指南
In Edexcel A-Level Mathematics, fluency with key formulae is essential for success across Pure Mathematics, Statistics and Mechanics. This guide brings together the most exam-relevant formulae and explains how to use them accurately. Remember that the formula booklet provides some results, but knowing these off by heart saves time and reduces errors.
在 Edexcel A-Level 数学中，熟练运用核心公式是纯数学、统计和力学取得高分的基础。本指南汇集了考试中最常见的重要公式，并解释如何准确使用。虽然公式手册会提供部分结果，但熟记这些公式可以节省时间、减少失误。
1. Quadratic Formula | 二次方程求根公式
For any quadratic equation ax² + bx + c = 0, where a ≠ 0, the roots are given by the quadratic formula:
x = (−b ± √(b² − 4ac)) / 2a
对于任意二次方程 ax² + bx + c = 0（a ≠ 0），其根由求根公式给出：x 等于 2a 分之 −b 加减根号下 b² − 4ac。
The discriminant Δ = b² − 4ac determines the nature of the roots: two real distinct roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0.
判别式 Δ = b² − 4ac 决定根的性质：若 Δ > 0，有两个不同实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。
A common mistake is to forget the minus sign on b or to divide only part of the numerator by 2a. Always write the entire numerator over the denominator before simplifying.
一个常见错误是忘记 b 的负号，或者只将分子的一部分除以 2a。化简前应始终将整个分子放在分母之上。
2. Laws of Indices and Surds | 指数与根式运算律
The laws of indices are essential for simplifying algebraic expressions and for calculus. The most important rules are:
指数运算法则是化简代数式以及微积分运算的基础。最重要的规则如下：
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁻ⁿ = 1 / aⁿ
- a^(1/n) = ⁿ√a
- a^(m/n) = ⁿ√(aᵐ)
Remember that a^(1/n) is the nth root of a, and a^(m/n) equals the nth root of aᵐ. Negative powers indicate reciprocals.
记住 a^(1/n) 表示 a 的 n 次方根，而 a^(m/n) 等于 aᵐ 的 n 次方根。负指数表示倒数。
When simplifying surds, use the rules √(ab) = √a × √b and √(a/b) = √a / √b for non-negative a, b. Rationalising the denominator often helps simplify an expression.
化简根式时，对于非负 a、b，可使用 √(ab) = √a × √b 和 √(a/b) = √a / √b。将分母有理化通常有助于化简表达式。
3. Logarithms | 对数运算
The logarithmic laws follow directly from index laws and are heavily tested in Pure Mathematics. For any base a > 0, a ≠ 1:
对数运算法则直接来源于指数法则，是纯数学的高频考点。对于任意底数 a > 0 且 a ≠ 1：
logₐ(xy) = logₐx + logₐy
logₐ(x/y) = logₐx − logₐy
logₐ(xⁿ) = n logₐx
The change of base formula is particularly useful when evaluating logarithms with a calculator: logₐb = log_c b / log_c a, where c is any valid base, often 10 or e.
换底公式在用计算器求值时尤其有用：logₐb = log_c b / log_c a，其中 c 为任意有效底数，通常取 10 或 e。
Do not confuse log(xy) with log x × log y; the correct identity is log(xy) = log x + log y. Also remember that logₐa = 1 and logₐ1 = 0.
不要混淆 log(xy) 与 log x × log y；正确恒等式为 log(xy) = log x + log y。同时记住 logₐa = 1 以及 logₐ1 = 0。
4. Trigonometric Identities | 三角恒等式
The Pythagorean identity and double-angle formulae are essential for solving trigonometric equations and for integration. The core identities are:
毕达哥拉斯恒等式与倍角公式在解三角方程和积分中必不可少。核心恒等式如下：
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
The double-angle formulae are particularly useful for transforming products of trigonometric functions into easier forms:
倍角公式在将三角函数的乘积转化为更简单的形式时特别有用：
sin(2θ) = 2sinθcosθ
cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
tan(2θ) = 2tanθ / (1 − tan²θ)
These identities allow you to rewrite expressions in forms that are easier to integrate or solve. For example, sin²θ can be written as (1 − cos 2θ) / 2.
这些恒等式可以把表达式改写成更容易积分或求解的形式。例如，sin²θ 可以写成 (1 − cos 2θ) / 2。
5. Differentiation Rules | 微分法则
You must know the standard derivatives and the chain, product and quotient rules. The main standard derivatives are shown in the table below.
你必须掌握常见函数的导数以及链式法则、乘积法则和商法则。下表列出了主要的常见函数导数。
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