📚 Year 10 Edexcel Maths Formula & Theorem Quick Reference Guide | Year 10 Edexcel 数学公式定理速查手册
This quick reference guide summarises the essential formulas and theorems for Year 10 Edexcel Mathematics. It covers number, algebra, geometry, statistics and more, helping you to review key concepts efficiently and prepare for assessments with confidence.
本速查手册总结了 Year 10 Edexcel 数学必备的公式和定理，涵盖数字、代数、几何、统计等内容，旨在帮助你高效复习核心知识点，自信应对各类评估。
1. Number Operations and Indices | 数字运算与指数定律
t of its prime factors, using a factor tree. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
质因数分解利用因子树将一个数表示为质因数的乘积，例如 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5。
The Highest Common Factor (HCF) is the largest number that divides exactly into two or more numbers, while the Lowest Common Multiple (LCM) is the smallest number that is a multiple of all of them.
最高公因数（HCF）是能同时整除两个或多个数的最大整数，最低公倍数（LCM）是这些数的倍数中最小的一个。
aᵐ × aⁿ = aᵐ⁺ⁿ (multiply powers: add exponents)
同底数幂相乘，指数相加。
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (divide powers: subtract exponents)
同底数幂相除，指数相减。
(aᵐ)ⁿ = aᵐⁿ (power of a power: multiply exponents)
幂的乘方，指数相乘。
a⁰ = 1 (any non‑zero number to the power zero equals 1)
任何非零数的零次方等于 1。
a⁻ⁿ = 1 / aⁿ (negative exponent gives a reciprocal)
负指数等于其正指数次方的倒数。
a½ = √a and a⅓ = ∛a (fractional exponents represent roots)
分数指数表示根式，a½ = √a，a⅓ = ∛a。
number as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. For instance, 4 500 = 4.5 × 10³.
标准形式将数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。例如 4 500 = 4.5 × 10³。
2. Algebra Essentials | 代数基础
ibutive law: a(b + c) = ab + ac. For double brackets, (x + a)(x + b) = x² + (a + b)x + ab.
展开括号运用分配律：a(b + c) = ab + ac。对于双括号，(x + a)(x + b) = x² + (a + b)x + ab。
6x + 9 = 3(2x + 3). Quadratic expressions like x² + 5x + 6 factorise into (x + 2)(x + 3).
因式分解是展开的逆运算。先提取公因式，如 6x + 9 = 3(2x + 3)。二次式 x² + 5x + 6 可分解为 (x + 2)(x + 3)。
Completing the square rewrites x² + bx in the form (x + p)² + q. For x² + 6x, half of 6 gives (x + 3)² − 9.
配方法将 x² + bx 写成 (x + p)² + q 的形式。对 x² + 6x，系数的一半为 3，得 (x + 3)² − 9。
ns rearranging to isolate a chosen variable. For v = u + at, solve for a: a = (v − u) / t.
公式变形指重新排列等式以求出指定变量。如 v = u + at 中，解出 a：a = (v − u) / t。
The quadratic formula: x = [ −b ± √b² − 4ac ] / (2a)
二次方程求根公式：x = [ −b ± √b² − 4ac ] / (2a)
3. Solving Equations and Inequalities | 解方程与不等式
operations on both sides until the unknown is isolated, e.g. 2x + 3 = 11 → 2x = 8 → x = 4.
解线性方程时，对等式两边执行逆运算，直至未知数单独出现，如 2x + 3 = 11 → 2x = 8 → x = 4。
c = 0 can be solved by factorising, completing the square, or using the quadratic formula.
形如 x² + bx + c = 0 的二次方程可通过因式分解、配方法或求根公式求解。
When solving inequalities, treat them like equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
解不等式时，可类似方程操作，但在乘或除以负数时，必须翻转不等号方向。
en circles ( < , > ) and closed circles ( ≤ , ≥ ), and list integer solutions if required.
在数轴上表示不等式时，开圈对应 < 或 >，闭圈对应 ≤ 或 ≥，并根据需要列举整数解。
Simultaneous equations can be solved by elimination (adding/subtracting equations) or by substitution. The intersection point of two lines gives the solution.
解联立方程组可使用消元法（加减方程）或代入法。两条直线的交点坐标即为方程组的解。
4. Sequences | 数列
consecutive terms. The nth term is given by uₙ = a + (n − 1)d, where a is the first term.
等差数列相邻两项的差 d 恒定。其第 n 项公式为 uₙ = a + (n − 1)d，其中 a 为首项。
has a constant ratio r between terms. Its nth term is uₙ = a × rⁿ⁻¹ (Year 10 higher tier).
等比数列相邻两项的比 r 恒定。第 n 项公式为 uₙ = a × rⁿ⁻¹（Year 10 高阶内容）。
Fibonacci‑type sequences: each term is the sum of the two preceding terms, e.g. 1, 1, 2, 3, 5, 8, … The rule is uₙ = uₙ₋₁ + uₙ₋₂.
类斐波那契数列中每一项为前两项之和，如 1, 1, 2, 3, 5, 8, … 其递归定义为 uₙ = uₙ₋₁ + uₙ₋₂。
A quadratic sequence has a second difference that is constant. The nth term contains an n² term and can be found by comparing with the sequence of square numbers.
二次数列的二阶差恒定。其通项公式包含 n² 项，可通过与平方数列对比求得。
The term‑to‑term rule describes how to move from one term to the next, e.g. ‘add 3’ or ‘multiply by 2’. Always write the position‑to‑term rule for the nth term where possible.
递推法则描述前后项之间的变化规律，如“加 3”或“乘以 2”。在可能的情况下应写出第 n 项的通项公式。
5. Graphs and Their Features | 图形及其特征
The equation of a straight line is y = mx + c, where m is the gradient and c is the y‑intercept. Parallel lines have the same gradient.
直线方程为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。平行直线斜率相等。
find the gradient between two points (x₁, y₁) and (x₂, y₂), use m = (y₂ − y₁) / (x₂ − x₁).
两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率用公式 m = (y₂ − y₁) / (x₂ − x₁) 计算。
ur where y = 0, and the turning point is found by completing the square or using symmetry.
二次函数图像 y = x² + bx + c 呈抛物线。图像与 x 轴的交点即方程的根，可通过配方法或对称性求得顶点。
Distance‑time graphs: a horizontal line means stationary; a straight sloping line represents constant speed. The gradient gives the speed.
距离‑时间图像中，水平线段表示静止，倾斜直线表示匀速运动，其斜率即为速度。
Velocity‑time graphs: the gradient represents acceleration, and the area under the graph represents the distance travelled. A horizontal line shows constant velocity.
速度‑时间图像中，斜率表示加速度，图像下方的面积代表行驶距离。水平线段表示速度恒定。
6. Ratio, Proportion and Rates | 比率、比例与变化率
Ratios compare quantities and can be simplified by dividing all parts by a common factor. The ratio 15:20 simplifies to 3:4.
比率用于比较两个量，可通过除以公因数化简，如 15:20 简化为 3:4。
To share a quantity in a given ratio, add the parts to find the total number of shares, then multiply accordingly. For example, share £60 in the ratio 2:3 gives £24 and £36.
按给定比率分配数量时，先将各项相加得到总份数，再按比例相乘。如将 £60 按 2:3 分配，得 £24 和 £36。
Direct proportion: y ∝ x means y = kx, where k is the constant of proportionality. As x doubles, y doubles.
正比例关系：y ∝ x 表示 y = kx，其中 k 为比例常数。x 加倍，y 也加倍。
Inverse proportion: y ∝ 1/x means y = k/x. As x increases, y decreases.
反比例关系：y ∝ 1/x 表示 y = k/x。x 增大时 y 减小。
Percentage change = (change / original) × 100%. Compound growth uses a multiplier, e.g. increasing £200 by 5% per year for 3 years: 200 × 1.05³.
百分比变化 = (变化量 / 原值) × 100%。复合增长使用乘数因子，例如每年增长 5%，£200 三年后为 200 × 1.05³。
ea. These are compound measures; rearranging them uses the same formula triangle approach.
速度 = 距离 / 时间，密度 = 质量 / 体积，压强 = 力 / 面积。这些复合单位可通过公式三角进行变形计算。
7. Geometry: Angles and Pythagoras | 几何：角度与勾股定理
Angles on a straight line add up to 180°, around a point 360°, and vertically opposite angles are equal.
直线上的角之和为 180°，一点周围的角之和为 360°，对顶角相等。
Angles in a triangle sum to 180°. In an isosceles triangle, base angles are equal. The exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形内角和为 180°。等腰三角形两底角相等。三角形的外角等于与它不相邻的两个内角之和。
Corresponding and alternate angles are equal when a transversal crosses parallel lines. Co‑interior angles sum to 180°.
平行线被一条横截线所截时，同位角和内错角分别相等，同旁内角之和为 180°。
The sum of interior angles of an n‑sided polygon is (n − 2) × 180°. Each exterior angle of a regular polygon is 360°/n.
n 边形的内角和为 (n − 2) × 180°。正多边形的每个外角等于 360°/n。
Pythagoras’ theorem: a² + b² = c², where c is the hypotenuse.
勾股定理：a² + b² = c²，其中 c 为斜边。
To find a shorter side, rearrange: a = √c² − b². Always check if the triangle is right‑angled before using the theorem.
求直角边时变形公式：a = √c² − b²。使用前务必确认三角形为直角三角形。
8. Trigonometry Basics | 三角学基础
In a right‑angled triangle, the three trigonometric ratios relate the angles to the side lengths:
在直角三角形中，三个三角比将角与边长联系起来：
sin θ = opposite / hypotenuse
sin θ = 对边 / 斜边
cos θ = adjacent / hypotenuse
cos θ = 邻边 / 斜边
tan θ = opposite / adjacent
tan θ = 对边 / 邻边
Use the mnemonic SOH CAH TOA to remember the ratios. To find a side, use sin, cos, or tan with the known angle and side. To find an angle, use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹.
用口诀 SOH CAH TOA 记忆。求边长时，利用已知角和边长选取正确的三角比；求角度时，使用反三角函数 sin⁻¹、cos⁻¹、tan⁻¹。
Label the sides relative to the given angle: the hypotenuse is the longest side, the opposite is the side facing the angle, and the adjacent is the remaining side.
标注边长时先确定角的对边、邻边和斜边——斜边是最长边，对边是角所对的边，邻边则是余下的那条边。
es for 0°, 30°, 45°, 60° and 90° should be memorised. For instance, sin 30° = ½, cos 45° = √2/2, tan 60° = √3.
需要熟记 0°、30°、45°、60° 和 90° 的精确三角值，例如 sin 30° = ½，cos 45° = √2/2，tan 60° = √3。
9. Area, Perimeter and Volume | 面积、周长与体积
Rectangle: area = l × w, perimeter = 2(l + w). Square: area = s², perimeter = 4s.
矩形：面积 = 长 × 宽，周长 = 2(长 + 宽)。正方形：面积 = 边长²，周长 = 4 × 边长。
× vertical height. Trapezium: area = ½ (a + b) × h, where a and b are the parallel sides.
三角形面积 = ½ × 底 × 高。平行四边形面积 = 底 × 垂直高度。梯形面积 = ½ (a + b) × h，其中 a 和 b 为两条平行边。
Prism volume = area of cross‑section × length. The surface area of a prism is the sum of the areas of all its
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