📚 Year 8 Edexcel Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 8 Edexcel 进阶数学：公式定理速查手册
This concise handbook brings together all the essential formulas, identities and geometric theorems covered in the Year 8 Edexcel Further Mathematics course. Each section pairs an explanation in English with its Chinese equivalent, making it an ideal revision aid for bilingual learners. Use it as a daily drill sheet or a last-minute check before tests.
本手册汇集了 Year 8 Edexcel 进阶数学课程中所有核心公式、恒等式与几何定理。每个小节都采用中英双语配对讲解，方便双语学生复习。可作为每日速练工具或考前快速查阅备忘录。
1. Algebraic Identities & Expansion | 代数恒等式与展开
The square of a binomial follows fixed patterns. For any terms a and b: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b². These are direct applications of the distributive law.
二项式的平方遵循固定模式。对任意项 a、b：(a + b)² = a² + 2ab + b²，(a – b)² = a² – 2ab + b²。两者均为分配律的直接应用。
The difference of two squares factorises neatly: a² – b² = (a + b)(a – b). This identity is very useful for simplifying algebraic fractions and solving quadratic-like equations.
平方差公式可以整齐地因式分解：a² – b² = (a + b)(a – b)。该恒等式对于化简代数分式以及解类二次方程非常有用。
The distributive law states that a(b + c) = ab + ac. Every term inside the bracket must be multiplied by the term outside. The same principle extends to double brackets: (a + b)(c + d) = ac + ad + bc + bd.
分配律指出 a(b + c) = ab + ac。括号内的每一项都要与括号外的项相乘。同样的原理也适用于双括号：(a + b)(c + d) = ac + ad + bc + bd。
2. Solving Linear Equations | 解线性方程
A linear equation is an equation of the form ax + b = c, where a, b and c are constants. To solve, perform inverse operations in the correct order: subtract or add first, then divide or multiply.
线性方程是形式为 ax + b = c 的方程，其中 a、b、c 为常数。求解时按照正确顺序进行逆运算：先加减，后乘除。
Example: Solve 3x + 5 = 11. Subtract 5 from both sides: 3x = 6. Divide by 3: x = 2. Always check by substituting the solution back into the original equation.
示例：解 3x + 5 = 11。两边同时减 5：3x = 6。除以 3：x = 2。记得将解代回原方程进行验证。
When the unknown appears on both sides, collect like terms onto one side first. For instance, 5x – 3 = 2x + 9 becomes 3x = 12, giving x = 4.
当未知数出现在等号两边时，先将同类项移到同一边。例如 5x – 3 = 2x + 9 化为 3x = 12，解得 x = 4。
3. Inequalities | 不等式
Inequalities use the symbols <, >, ≤ and ≥. Solving them is similar to solving equations, but one key rule: if you multiply or divide both sides by a negative number, the inequality sign must be reversed.
不等式使用符号 <、>、≤ 和 ≥。解不等式的方法与解方程类似，但有一条关键规则：若两边同乘或同除一个负数，不等式方向必须反转。
Example: –2x ≤ 8. Divide both sides by –2: x ≥ –4. The ≤ sign becomes ≥ because of the division by a negative.
示例：–2x ≤ 8。两边除以 –2：x ≥ –4。由于除以负数，≤ 号变成 ≥。
Solutions can be shown on a number line with open circles for strict inequalities (< or >) and closed circles for inclusive ones (≤ or ≥). The arrow indicates the range of values.
解集可在数轴上表示：严格不等式（< 或 >）用空心圆，包含等号的不等式（≤ 或 ≥）用实心圆。箭头表示取值范围。
4. Sequences & the nth Term | 数列与第 n 项
A linear sequence adds (or subtracts) the same amount each time. The rule for the nth term is nth term = dn + (a – d), where d is the common difference and a is the first term.
线性数列每次加上（或减去）相同的量。第 n 项的通项公式为 第 n 项 = dn + (a – d)，其中 d 是公差，a 是首项。
Example: Sequence 5, 9, 13, 17, … has d = 4 and a = 5. So the nth term is 4n + 1. To find the 10th term, substitute n = 10: 4 × 10 + 1 = 41.
示例：数列 5, 9, 13, 17, … 公差 d = 4，首项 a = 5。因此通项为 4n + 1。求第 10 项时，代入 n = 10：4 × 10 + 1 = 41。
If the sequence descends, the common difference is negative. The sequence 20, 17, 14, 11, … has d = –3, yielding nth term = –3n + 23.
若数列递减，则公差为负数。数列 20, 17, 14, 11, … 的公差 d = –3，通项为 –3n + 23。
5. Linear Graphs | 直线图
The equation of a straight line is written as y = mx + c. Here m is the gradient (steepness) and c is the y-intercept (the point where the line crosses the y-axis).
直线方程写作 y = mx + c。其中 m 表示斜率（倾斜程度），c 是 y 轴截距（直线与 y 轴的交点）。
The gradient is calculated using two points: m = (y₂ – y₁) / (x₂ – x₁). A positive m means the line slopes upward; a negative m slopes downward. Horizontal lines have m = 0, vertical lines have an undefined gradient.
斜率由两点坐标计算：m = (y₂ – y₁) / (x₂ – x₁)。m 为正时直线向上倾斜，为负时向下倾斜。水平线的斜率是 0，铅垂线的斜率无定义。
To plot a linear graph, find three points that satisfy the equation, plot them and draw a straight line through them. The y-intercept method starts at (0, c) and uses the gradient to find a second point.
绘制直线图时，找出满足方程的三个点并过点画直线。截距法从 (0, c) 开始，再利用斜率找到第二个点。
6. Pythagoras & Trigonometry | 勾股定理与三角
In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c², where c is the longest side opposite the right angle.
在任何直角三角形中，斜边的平方等于两直角边的平方和：a² + b² = c²，其中 c 是直角所对的最长边。
Basic trigonometric ratios link angles to side lengths: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. The mnemonic SOH CAH TOA helps memorise them.
基本三角比连接了角与边长：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。口诀 SOH CAH TOA 有助于记忆。
Pythagoras can be used to find a missing side, and trigonometry to find a missing angle using inverse functions. For example, if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°.
勾股定理可用来求边长，三角比配合反函数可用来求未知角。例如 sin θ = 0.5，则 θ = sin⁻¹(0.5) = 30°。
7. Perimeter, Area & Volume | 周长、面积与体积
Key area formulas: Rectangle = l × w; Triangle = ½ × base × height; Parallelogram = base × height; Trapezium = ½ (a + b)h, where a and b are parallel sides; Circle = πr², and circumference = 2πr.
核心面积公式：矩形 = 长 × 宽；三角形 = ½ × 底 × 高；平行四边形 = 底 × 高；梯形 = ½ (a + b)h，其中 a 和 b 是平行边；圆面积 = πr²，周长 = 2πr。
Volume of a prism is found by multiplying the area of its cross-section by its length. For a cuboid, volume = l × w × h. For a cylinder, volume = πr²h.
棱柱的体积等于横截面积乘以长度。长方体体积 = 长 × 宽 × 高。圆柱体积 = πr²h。
Surface area is the total area of all faces. For a cuboid, it is 2(lw + lh + wh). For a cylinder, it includes two circles and a curved rectangle: 2πr² + 2πrh.
表面积是所有面的总面积。长方体表面积 = 2(lw + lh + wh)。圆柱表面积包括两个圆和一个弯曲矩形：2πr² + 2πrh。
8. Indices & Standard Form | 指数与标准形式
The laws of indices simplify expressions involving powers: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ. Any non-zero number to the power 0 equals 1: a⁰ = 1.
指数法则可化简幂运算表达式：aᵐ × aⁿ = aᵐ⁺ⁿ；aᵐ ÷ aⁿ = aᵐ⁻ⁿ；(aᵐ)ⁿ = aᵐⁿ。任何非零数的 0 次幂等于 1：a⁰ = 1。
Negative indices indicate reciprocals: a⁻ⁿ = 1 / aⁿ. For example, 2⁻³ = 1 / 2³ = 1/8. Fractional indices represent roots: a^(1/n) = the nth root of a.
负指数表示倒数：a⁻ⁿ = 1 / aⁿ。例如 2⁻³ = 1 / 2³ = 1/8。分数指数表示开方：a^(1/n) = a 的 n 次方根。
Standard form writes a number as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. It is useful for very large or very small numbers, e.g. 5.6 × 10⁴ = 56 000, and 3.2 × 10⁻³ = 0.0032.
标准形式将数字写作 A × 10ⁿ，其中 1 ≤ A < 10 且 n 为整数。它适用于很大或很小的数，如 5.6 × 10⁴ = 56 000，3.2 × 10⁻³ = 0.0032。
9. Statistics & Probability | 统计与概率
The mean is calculated by summing all data values and dividing by the number of values: Mean = sum of data ÷ number of data points. The median is the middle value when data are ordered; the mode is the most frequent value; the range is the difference between the largest and smallest.
平均数是将所有数据值相加再除以数据个数：平均数 = 数据总和 ÷ 数据个数。中位数是排序后中间的值；众数是出现次数最多的值；极差是最大值与最小值的差。
Probability measures how likely an event is: P(Event) = number of favourable outcomes / total number of possible outcomes. Probabilities lie between 0 and 1, and the sum of probabilities of all possible outcomes is 1.
概率度量事件发生的可能性：P(事件) = 有利结果数 / 所有可能结果数。概率值介于 0 到 1 之间，且所有可能结果的概率之和为 1。
For mutually exclusive events, P(A or B) = P(A) + P(B). The probability of an event not happening is 1 – P(event). Tree diagrams and sample space diagrams help organise outcomes.
对于互斥事件，P(A 或 B) = P(A) + P(B)。事件不发生的概率是 1 – P(事件)。树形图和样本空间图有助于梳理所有可能结果。
10. Ratio, Proportion & Rates | 比、比例与速率
A ratio compares quantities of the same kind. Ratios can be simplified like fractions and are often written as a : b. When dividing a quantity in a given ratio, find the total number of parts first.
比用来比较同类量。比可以像分数一样化简，常写作 a : b。当按给定比例分配总量时，首先求出总份数。
Direct proportion means two quantities increase together. It can be expressed as y = kx, where k is the constant of proportionality. The graph of direct proportion is a straight line through the origin.
正比例表示两个量同时增加，可表示为 y = kx，其中 k 是比例常数。正比例图像是一条过原点的直线。
Inverse proportion means that as one quantity increases, the other decreases. It follows y = k / x and produces a curved graph called a hyperbola. The product xy remains constant.
反比例表示一个量增加时另一个量减少，遵循 y = k / x，图像是一条称为双曲线的曲线。乘积 xy 保持恒定。
Common rate formulas: Speed = distance ÷ time; Density = mass ÷ volume; Unit price = total cost ÷ number of items. Always ensure consistent units when calculating rates.
常见速率公式：速度 = 路程 ÷ 时间；密度 = 质量 ÷ 体积；单价 = 总价 ÷ 件数。计算速率时务必保持单位一致。
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