📚 Year 9 AQA Further Maths: Quick Reference Handbook of Formulas and Theorems | Year 9 AQA 进阶数学：公式定理速查手册
This handbook provides a concise yet comprehensive collection of the key formulas, rules and theorems you will encounter in Year 9 AQA Further Mathematics. Use it as a daily companion to speed up revision, check your working and build confidence before assessments.
本手册为学习 AQA 进阶数学的 Year 9 学生梳理了必备公式、定理与法则，涵盖代数、三角、微积分、矩阵等核心内容，适合作为日常速查工具，帮助巩固知识并提升备考效率。
1. Algebraic Fundamentals: Indices, Expanding and Factorising | 代数基础：指数、展开与因式分解
For any non-zero number a and integers m, n, the laws of indices are am × an = am+n, am ÷ an = am−n, (am)n = amn, a0 = 1 and a−n = 1/an.
对于任意非零实数 a 和整数 m, n：同底数幂相乘指数相加，相除指数相减，幂的乘方指数相乘，a0=1，a−n = 1/an。
Fractional indices link powers and roots: a1/n = n√a, and am/n = n√am = (n√a)m.
分数指数连接幂与根式：a1/n 表示 a 的 n 次方根，am/n 等于 a 的 m 次幂再开 n 次方根。
b²; (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b²; (x + a)(x + b) = x² + (a+b)x + ab.
重要展开公式：完全平方、平方差以及二次项系数为1的十字相乘展开。
iply to give ac and add to give b (when a=1, simply two numbers with product c and sum b).
因式分解是展开的逆过程：提取公因式、使用平方差公式，或对二次三项式寻找满足乘积为 c、和为 b 的两个数（当二次项系数为1时）。
2. Surds and Rationalising Denominators | 根式与分母有理化
A surd is an irrational root, e.g. √2 or ³√5. Simplify surds using √ab = √a × √b and √a²b = a√b for a ≥ 0.
根式是无理数的根号形式。简化根式时利用 √ab = √a · √b，并将完全平方因子移出根号，如 √a²b = a√b。
To rationalise a denominator containing a single surd, multiply numerator and denominator by that surd: 1/√a = √a / a.
分母含单项根式时，分子分母同乘该根式即可：1/√a = √a / a。
If the denominator is of the form a + √b, multiply top and bottom by its conjugate a − √b, using (a + √b)(a − √b) = a² − b to remove the surd.
若分母形如 a+√b，分子分母同乘共轭根式 a−√b，借助平方差消去根号：(a+√b)(a−√b)=a²−b。
3. Quadratic Equations | 二次方程
0). Solutions can be found by factorising, completing the square or the quadratic formula.
二次方程一般式为 ax² + bx + c = 0 (a ≠ 0)。可用因式分解、配方法或求根公式求解。
The quadratic formula gives the roots: x = (−b ± √b² − 4ac) / (2a). The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 ⇒ two distinct real roots; Δ = 0 ⇒ one repeated real root; Δ < 0 ⇒ no real roots.
求根公式为 x = (−b ± √b² − 4ac) / (2a)。判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 两个不等实根；Δ = 0 一个重根；Δ < 0 无实根。
Completing the square rewrites x² + bx as (x + b/2)² − (b/2)². The vertex form is a(x − h)² + k with vertex (h, k).
配方法将 x² + bx 变形为 (x + b/2)² − (b/2)²。二次函数顶点式为 a(x−h)² + k，顶点为 (h, k)。
4. Simultaneous Equations and Inequalities | 联立方程与不等式
To solve linear simultaneous equations, use elimination (add or subtract to remove one variable) or substitution (express one variable in terms of the other). Always check your solutions in both original equations.
解线性方程组时，使用消元法（加减消去一个变量）或代入法（将一个变量用另一个表示）。务必把解代入原方程组检验。
Linear inequalities are solved similarly to equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
一元一次不等式的解法与方程类似，但两边同乘或同除一个负数时，不等号方向要改变。
Quadratic inequalities (e.g. x² − 5x + 6 > 0) are solved by finding critical values from the associated equation, sketching the parabola and selecting intervals where the graph is above or below the x‑axis.
解二次不等式时，先求对应方程的根，画出抛物线草图，再根据开口方向选取满足大于零或小于零的区间。
5. Polynomials and the Factor Theorem | 多项式与因式定理
A polynomial f(x) can be divided by (x − a) using long division or synthetic division. The Remainder Theorem states that the remainder is f(a).
多项式 f(x) 除以 (x−a) 可采用长除法或综合除法。余数定理指出，余数等于 f(a)。
x − a) is a factor of f(x) if and only if f(a) = 0. This helps factorise cubics and higher‑degree polynomials.
因式定理：f(a)=0 当且仅当 (x−a) 是 f(x) 的一个因式。该定理可用来分解三次及更高次多项式。
To factorise, find an integer root by testing factors of the constant term, then divide to obtain a quadratic factor.
分解时，先通过检验常数项的因子找到一个整数根，再用除法得到二次因式，继续分解。
6. Binomial Expansion | 二项式展开
nteger n, the binomial expansion of (a + b)ⁿ is given by the sum from k=0 to n of C(n,k) an−k bk, where the binomial coefficient C(n,k) = n! / [k!(n−k)!].
当 n 为正整数时，(a+b)ⁿ 的二项展开式可表示为 Σ C(n,k) an−k bk（k从0到n）。其中二项式系数 C(n,k) = n! / [k!(n−k)!]。
Pascal' in coefficients. For small powers: (a+b)² = a² + 2ab + b², (a+b)³ = a³ + 3a²b + 3ab² + b³.
帕斯卡三角形可快速获取系数。常用展开：(a+b)² = a²+2ab+b²，(a+b)³ = a³+3a²b+3ab²+b³。
The expansion of (1 + x)ⁿ for small n follows the same pattern and is particularly useful for approximations when |x| is small.
(1+x)ⁿ 的展开同样遵循系数规律，在 |x| 很小时可用于近似计算。
7. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆
(x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). A line with equation y = mx + c has gradient m and y‑intercept c.
过两点 (x₁,y₁) 和 (x₂,y₂) 的直线斜率为 m = (y₂−y₁)/(x₂−x₁)。斜截式 y = mx + c 中，m 为斜率，c 为 y 轴截距。
ed neither is vertical). The equation of a line can also be written as y − y₁ = m(x − x₁).
平行直线斜率相等；互相垂直的直线斜率满足 m₁·m₂ = −1。点斜式为 y − y₁ = m(x − x₁)。
(y − b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √g² + f² − c.
圆心为 (a,b)、半径为 r 的圆的方程为 (x−a)²+(y−b)² = r²。一般式 x²+y²+2gx+2fy+c=0 中，圆心为 (−g,−f)，半径 √g²+f²−c。
8. Trigonometry | 三角学
In right‑ opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH CAH TOA).
直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Pythagorean identity: sin²θ + cos²θ = 1. Also, tan θ = sin θ / cos θ.
恒等式：sin²θ + cos²θ = 1，且 tan θ = sin θ / cos θ。
a / sin A = b / sin B = c / sin C (= 2R). Use it to find sides or angles in any triangle.
正弦定理：a/sin A = b/sin B = c/sin C（等于外接圆直径 2R）。
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