📚 KS3 CCEA Further Mathematics: Quick Reference Handbook of Formulas and Theorems | KS3 CCEA 进阶数学：公式定理速查手册
This handbook provides a concise, exam-focused summary of the essential formulas and theorems you will encounter in KS3 CCEA Further Mathematics. Use it to reinforce your understanding, speed up homework, and prepare confidently for assessments. Every section pairs core content in English with its Chinese equivalent, ensuring clarity for bilingual learners.
本手册为 KS3 CCEA 进阶数学提供了简明、贴近考试的必背公式与定理总结。用它来巩固理解、提升作业效率、自信备考。每个小节都采用中英双语对照，确保双语学习者能够清晰掌握要点。
1. Algebraic Expressions and Operations | 代数表达式与运算
When expanding double brackets, multiply each term in the first bracket by every term in the second: (a + b)(c + d) = ac + ad + bc + bd.
展开双括号时，将第一个括号中的每一项分别乘以第二个括号中的每一项：(a + b)(c + d) = ac + ad + bc + bd。
Memorise these three special products – they appear constantly in factorisation and simplification.
牢记下面三个特殊乘积——它们在因式分解与化简中频繁出现。
| Perfect square (sum) | (a + b)² = a² + 2ab + b² |
| Perfect square (difference) | (a − b)² = a² − 2ab + b² |
| Difference of two squares | (a + b)(a − b) = a² − b² |
完全平方（和）：(a + b)² = a² + 2ab + b²；完全平方（差）：(a − b)² = a² − 2ab + b²；平方差公式：(a + b)(a − b) = a² − b²。
Factorising is the reverse process: look for a common factor first, then recognise these standard forms when possible.
因式分解是展开的逆过程：先提取公因式，再尽量套用上述标准形式。
2. Laws of Indices | 指数定律
Indices rules simplify expressions involving powers. They apply when the base is the same.
指数定律可化简幂的运算，前提是底数相同。
- Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ
乘法定律：aᵐ × aⁿ = aᵐ⁺ⁿ - Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)
除法定律：aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0) - Power of a power: (aᵐ)ⁿ = aᵐⁿ
幂的乘方：(aᵐ)ⁿ = aᵐⁿ - Zero index: a⁰ = 1 (a ≠ 0)
零指数：a⁰ = 1 (a ≠ 0) - Negative index: a⁻ⁿ = 1 / aⁿ
负指数：a⁻ⁿ = 1 / aⁿ
Fractional indices link to roots: a1/2 = √a and a1/n = n√a. These are particularly useful in advanced algebra.
分数指数与根式关联：a1/2 = √a，a1/n = n√a。这在进阶代数中尤其有用。
3. Linear Equations and Inequalities | 线性方程与不等式
To solve a linear equation ax + b = c, isolate x by performing inverse operations: subtract b, then divide by a.
解线性方程 ax + b = c 时，通过逆运算分离 x：先减 b，再除以 a。
For linear inequalities, the same steps work, but reverse the inequality sign when multiplying or dividing by a negative number.
解线性不等式时步骤相同，但当乘以或除以一个负数时，必须反转不等号方向。
Example: Solve −2x > 6. Divide by −2 to get x < −3.
例子：解 −2x > 6。两边除以 −2 得 x < −3。
r line in exams, using open circles for strict inequalities and closed circles for ≤ or ≥.
考试时一定要在数轴上表示解集：严格不等号用空心圆，≤ 或 ≥ 用实心圆。
4. Quadratic Equations | 二次方程
A quadratic equation has the general form ax² + bx + c = 0, where a ≠ 0. The most common solution methods are factorising, completing the square, and the quadratic formula.
二次方程的一般形式为 ax² + bx + c = 0（a ≠ 0）。最常见的解法包括因式分解、配方法和求根公式。
Quadratic Formula: x = [ −b ± √b² − 4ac ] / (2a)
求根公式：x = [ −b ± √b² − 4ac ] / (2a)
The discriminant Δ = b² − 4ac ves two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
判别式 Δ = b² − 4ac 决定了根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实数根。
Always set the equation to zero before applying any method.
无论用哪种方法，都需先将方程化为等于零的形式。
5. Pythagoras' Theorem and Trigonometry | 勾股定理与三角学
In any right‑angled triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides:
在任何直角三角形中，斜边 c 的平方等于两直角边的平方和：
a² + b² = c²
The three trigonometric ratios for an acute angle θ in a right‑angled triangle are defined as:
直角三角形中锐角 θ 的三个三角比定义为：
| sin θ | opposite / hypotenuse | 正弦：对边 / 斜边 |
| cos θ | adjacent / hypotenuse | 余弦：邻边 / 斜边 |
| tan θ | opposite / adjacent | 正切：对边 / 邻边 |
Learn the exact values for 0°, 30°, 45°, 60° and 90° – they are often tested without a calculator.
牢记 0°、30°、45°、60° 和 90° 的精确值——考试中经常不允许使用计算器。
6. Geometry Formulas | 几何公式
These formulas are essential for area and perimeter calculations. All linear dimensions must be in the same units.
以下公式是面积和周长计算的基础。所有线性尺寸必须使用相同单位。
- Triangle: Area = ½ × base × vertical height
三角形：面积 = ½ × 底 × 垂直高 - Rectangle: Area = length × width, Perimeter = 2(l + w)
矩形：面积 = 长 × 宽，周长 = 2(长 + 宽) - Parallelogram: Area = base × perpendicular height
平行四边形：面积 = 底 × 垂直高 - Trapezium: Area = ½(a + b)h, where a and b are parallel sides and h is perpendicular height
梯形：面积 = ½(a + b)h，其中 a、b 为平行边，h 为高 - Circle: Circumference = 2πr, Area = πr²
圆：周长 = 2πr，面积 = πr²
Always use the perpendicular height, not slant height, in triangle and parallelogram areas.
计算三角形和平行四边形面积时务必使用垂直高，而不是斜高。
7. Volume and Surface Area | 体积与表面积
For 3D shapes, volume measures capacity and surface area covers all faces. Prisms are particularly important.
对于三维图形，体积衡量容量，表面积覆盖所有面。棱柱的体积公式尤其重要。
- Cuboid: Volume = l × w × h; Surface area = 2(lw + lh + wh)
长方体：体积 = 长 × 宽 × 高；表面积 = 2(lw + lh + wh) - Prism: Volume = area of cross‑section × length
棱柱：体积 = 横截面积 × 长度 - Cylinder: Volume = πr²h; Curved surface area = 2πrh; Total surface area = 2πr(h + r)
圆柱体：体积 = πr²h；侧面积 = 2πrh；总表面积 = 2πr(h + r)
Surface area of a prism is the sum of the areas of all its faces. For a cylinder, remember to include the two circular ends.
棱柱的表面积是所有面的面积之和。计算圆柱体时，不要忘记加上两个圆形底面。
8. Statistics and Probability | 统计与概率
Descriptive statistics summarise data. The three measures of central tendency are the mean, median and mode.
描述性统计用于概括数据。三种集中趋势度量是平均数、中位数和众数。
- Mean: sum of all values ÷ number of values
平均数：所有数值之和 ÷ 数值的个数 - Median: middle value when data is ordered
中位数：数据排序后位于中间的值 - Mode: most frequently occurring value
众数：出现次数最多的值 - Range: maximum − minimum
极差：最大值 − 最小值
Probability of an event E is P(E) = Number of favourable outcomes / Total number of possible outcomes, provided all outcomes are equally likely.
若所有结果等可能，事件 E 的概率为 P(E) = 有利结果数 / 可能结果总数。
Probabilities always lie between 0 and 1. The sum of probabilities of all mutually exclusive outcomes is 1.
概率值始终介于 0 和 1 之间。所有互斥结果的概率之和为 1。
9. Sequences | 数列
A sequence is a list of numbers following a rule. The nth term allows you to find any term directly.
数列是按规律排列的一列数。第 n 项公式能直接求出任意位置的项。
- Arithmetic sequence: nth term = a + (n − 1)d, where a is the first term and d is the common difference.
等差数列：第 n 项 = a + (n − 1)d，a 为首项，d 为公差。 - Quadratic sequence: second difference constant; nth term has the form an² + bn + c. Find a, b, c by comparing terms.
二次数列：二次差为常数；第 n 项形式为 an² + bn + c。通过对比项求出 a、b、c。
To determine whether a sequence is arithmetic, subtract consecutive terms. For quadratic, examine second differences.
判断是否为等差数列：连续两项相减。判断是否为二次数列：检查二次差。
10. Coordinate Geometry | 坐标几何
Coordinate geometry links algebraic equations to geometric points on a plane. The key tools are distance, midpoint and gradient.
坐标几何将代数方程与平面上的几何点联系起来。核心工具是距离、中点和斜率。
- Distance between two points (x₁, y₁) and (x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²]
两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²] - Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
中点：((x₁ + x₂)/2, (y₁ + y₂)/2) - Gradient (slope): m = (y₂ − y₁)/(x₂ − x₁)
斜率：m = (y₂ − y₁)/(x₂ − x₁)
The equation of a straight line is often written as y = mx + c, where m is the gradient and c is the y‑intercept. Parallel lines have the same gradient.
直线方程常写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距。平行直线斜率相等。
You can also use the point‑gradient form: y − y₁ = m(x − x₁) to find the equation.
也可使用点斜式求方程：y − y₁ = m(x − x₁)。
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