📚 GCSE Edexcel Further Maths: Formula & Theorem Quick Reference Handbook | GCSE Edexcel 进阶数学：公式定理速查手册
This concise handbook collates the essential formulas, theorems, and identities required for the Edexcel GCSE Further Mathematics specification. Use it for rapid revision, consolidation, and as a pre-exam checklist. Mastery of these formulas will strengthen your problem-solving across Algebra, Calculus, Trigonometry, Matrices, Vectors, and more.
这本精简手册汇编了 Edexcel GCSE 进阶数学考试所需的关键公式、定理和恒等式。它可用于快速复习、巩固知识，同时也可作为考前核对清单。熟练掌握这些公式将提升你在代数、微积分、三角学、矩阵、向量等多个领域的解题能力。
1. Algebraic Essentials | 代数基础
For a quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula: x = [−b ± √(b² − 4ac)] / (2a). The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots.
对于二次方程 ax² + bx + c = 0，求解公式为 x = [−b ± √(b² − 4ac)] / (2a)。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 无实根。
The Factor Theorem states that (x − a) is a factor of a polynomial f(x) if and only if f(a) = 0. The Remainder Theorem tells us that when f(x) is divided by (x − a), the remainder is f(a).
因式定理指出，(x − a) 是多项式 f(x) 的因式当且仅当 f(a) = 0。余数定理告诉我们，当 f(x) 除以 (x − a) 时，余数为 f(a)。
Completing the square for ax² + bx + c transforms the expression into a(x + p)² + q, where p = b/(2a) and q = c − (b²/(4a)). This is essential for finding the vertex of a parabola.
配方法将 ax² + bx + c 转化为 a(x + p)² + q，其中 p = b/(2a)，q = c − (b²/(4a))。这在求抛物线的顶点时非常重要。
| Concept / 概念 | Formula / 公式 |
|---|---|
| Quadratic Formula / 二次公式 | x = [−b ± √(b² − 4ac)] / (2a) |
| Discriminant / 判别式 | Δ = b² − 4ac |
| Factor Theorem / 因式定理 | f(a) = 0 ⇔ (x − a) is a factor |
| Completing square / 配方法 | ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a)) |
2. Coordinate Geometry | 坐标几何
The distance between two points (x₁, y₁) and (x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]. The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). The gradient of the line through these points is m = (y₂ − y₁)/(x₂ − x₁).
两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离为 d = √[(x₂ − x₁)² + (y₂ − y₁)²]。中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。过这两点的直线斜率为 m = (y₂ − y₁)/(x₂ − x₁)。
The equation of a straight line can be written as y = mx + c (gradient-intercept form) or y − y₁ = m(x − x₁) (point-gradient form). Two lines are parallel if m₁ = m₂; they are perpendicular if m₁ × m₂ = −1.
直线方程可写成 y = mx + c (斜截式) 或 y − y₁ = m(x − x₁) (点斜式)。两直线平行，则 m₁ = m₂；两直线垂直，则 m₁ × m₂ = −1。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². To find the centre and radius from x² + y² + 2gx + 2fy + c = 0, complete the square to get 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。从一般式 x² + y² + 2gx + 2fy + c = 0 求圆心和半径时，用配方法得圆心 (−g, −f)，半径 √(g² + f² − c)。
Gradient from two points / 两点求斜率: m = (y₂ − y₁)/(x₂ − x₁)
3. Calculus: Differentiation and Integration | 微积分：微分与积分
The derivative of xⁿ is nxⁿ⁻¹ for any rational n. For a linear combination, d/dx [k·f(x)] = k·f'(x). The derivative of a constant is zero. The second derivative d²y/dx² indicates concavity.
xⁿ 的导数为 nxⁿ⁻¹（ n 为任意有理数）。对于线性组合，有 d/dx [k·f(x)] = k·f'(x)。常数的导数为零。二阶导数 d²y/dx² 用于判断凹凸性。
Stationary points occur where dy/dx = 0. Use the first derivative test or the second derivative to classify: if d²y/dx² > 0, it’s a minimum; if < 0, it's a maximum; if = 0, check sign change.
驻点出现在 dy/dx = 0 处。用一阶导数检验或二阶导数进行分类：若 d²y/dx² > 0，则为极小值点；若 < 0，则为极大值点；若 = 0，需检查符号变化。
Integration is the reverse of differentiation: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, for n ≠ −1. When n = −1, ∫ x⁻¹ dx = ln|x| + C. Definite integrals give the area between the curve and the x-axis: ∫ₐᵇ f(x) dx.
积分是微分的逆运算：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，(n ≠ −1)。当 n = −1 时，∫ x⁻¹ dx = ln|x| + C。定积分用于求曲线与 x 轴之间的面积：∫ₐᵇ f(x) dx。
The area between two curves y = f(x) (upper) and y = g(x) (lower) from a to b is ∫ₐᵇ [f(x) − g(x)] dx. Always sketch first to identify the top and bottom curves.
两条曲线 y = f(x) （上）和 y = g(x) （下）在区间 [a, b] 之间的面积为 ∫ₐᵇ [f(x) − g(x)] dx。务必先画草图以确定上下曲线。
| Function / 函数 | Derivative / 导数 | Integral / 不定积分 |
|---|---|---|
| xⁿ (n ≠ −1) | nxⁿ⁻¹ | xⁿ⁺¹/(n+1) + C |
| eˣ | eˣ | eˣ + C |
| ln x | 1/x | x ln x − x + C |
4. Trigonometry | 三角学
In right-angled triangles, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The exact values for 30°, 45°, 60° must be memorised, and angles are often expressed in radians: π rad = 180°.
在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。必须熟记 30°、45°、60° 的精确值，角度常以弧度表示：π 弧度 = 180°。
The sine rule: a / sin A = b / sin B = c / sin C, used for non-right-angled triangles when given two angles and a side or two sides and a non-included angle. The cosine rule: a² = b² + c² − 2bc cos A, used for two sides and the included angle or three sides.
正弦定理：a / sin A = b / sin B = c / sin C，适用于非直角三角形，已知两角一边或两边一对角。余弦定理：a² = b² + c² − 2bc cos A，适用于已知两边及其夹角或已知三边。
Key identities: sin² θ + cos² θ = 1, tan θ = sin θ / cos θ. Area of a triangle: ½ ab sin C. The radian measure formula: arc length s = rθ, sector area A = ½ r²θ.
关键恒等式：sin² θ + cos² θ = 1，tan θ = sin θ / cos θ。三角形面积：½ ab sin C。弧度制公式：弧长 s = rθ，扇形面积 A = ½ r²θ。
Sine Rule / 正弦定理: a/sin A = b/sin B = c/sin C
Cosine Rule / 余弦定理: a² = b² + c² − 2bc cos A
5. Matrices | 矩阵
A matrix is a rectangular array of numbers. Two matrices can be added or subtracted if they have the same dimensions. Multiplication of matrices: for A (m × n) and B (n × p), the product AB is m × p, with entry (i, j) equal to the dot product of the ith row of A and the jth column of B.
矩阵是一个矩形数字阵列。维度相同的两个矩阵可以相加或相减。矩阵乘法：对于 A (m × n) 和 B (n × p)，乘积 AB 的维度为 m × p，其第 (i, j) 元等于 A 的第 i 行与 B 的第 j 列的点积。
The determinant of a 2 × 2 matrix A = [a b; c d] is det A = ad − bc. A matrix is singular if det = 0 and non-singular otherwise. The inverse is A⁻¹ = (1/det) [d −b; −c a] provided det ≠ 0.
二阶矩阵 A = [a b; c d] 的行列式为 det A = ad − bc。若行列式为零，矩阵奇异；否则非奇异。逆矩阵为 A⁻¹ = (1/det) [d −b; −c a]，要求 det ≠ 0。
Matrices represent linear transformations: rotations, reflections, enlargements. For example, a rotation of 90° anticlockwise about the origin is represented by [0 −1; 1 0]. The image of a point (x, y) is obtained by multiplying the matrix by the column vector.
矩阵可表示线性变换：旋转、反射、放大。例如，绕原点逆时针旋转 90° 的变换矩阵为 [0 −1; 1 0]。点 (x, y) 的像通过矩阵乘以列向量得到。
Solving simultaneous linear equations using matrices: AX = B ⇒ X = A⁻¹B. The identity matrix I = [1 0; 0 1] leaves any matrix unchanged when multiplied.
用矩阵解联立线性方程组：AX = B ⇒ X = A⁻¹B。单位矩阵 I = [1 0; 0 1] 与任何矩阵相乘均不改变其值。
6. Vectors | 向量
A vector has magnitude and direction. It can be written in column form v = (x, y) or as v = xi + yj. The magnitude |v| = √(x² + y²). A unit vector in the same direction is v/|v|.
向量有大小和方向。它可用列向量 (x, y) 表示，或表示为 xi + yj。模长 |v| = √(x² + y²)。同方向的单位向量为 v/|v|。
Vector addition: a + b = (a₁ + b₁, a₂ + b₂). Scalar multiplication: ka = (ka₁, ka₂). The vector from A to B is AB = b − a. The midpoint of AB has position vector ( a + b )/2.
向量加法：a + b = (a₁ + b₁, a₂ + b₂)。标量乘法：ka = (ka₁, ka₂)。从 A 到 B 的向量为 AB = b − a。AB 的中点的位置向量为 ( a + b )/2。
The dot product a · b = a₁b₁ + a₂b₂ = |a||b| cos θ, where θ is the angle between them. Two vectors are perpendicular if their dot product is zero. For parallel vectors, a = kb.
点积 a · b = a₁b₁ + a₂b₂ = |a||b| cos θ，其中 θ 为两向量的夹角。若点积为零，则两向量垂直。若 a = kb，则两向量平行。
The vector equation of a straight line passing through point A with direction vector d is r = a + td, where t is a scalar parameter. This is used to find intersections and positions on the line.
过点 A 且方向向量为 d 的直线的向量方程为 r = a + td，其中 t 为标量参数。该方程用于求直线上的交点和位置。
7. Sequences and Series | 数列与级数
An arithmetic sequence has a common difference d: uₙ = a + (n−1)d. The sum to n terms is Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term.
等差数列有公差 d，通项为 uₙ = a + (n−1)d。前 n 项和为 Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l)，其中 l 为末项。
A geometric sequence has a common ratio r (|r| ≠ 1 for sum): uₙ = arⁿ⁻¹. The sum to n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. If |r| < 1, the sum to infinity exists: S∞ = a/(1 − r).
等比数列有公比 r (|r| ≠ 1 时求和)，通项为 uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r) (r ≠ 1)。若 |r| < 1，则存在无穷和：S∞ = a/(1 − r)。
Be careful to distinguish between sequences defined by a term-to-term rule (recurrence) and a position-to-term rule (nth term formula). Both types appear in GCSE Further Maths.
注意区分由递推关系（逐项规则）定义的数列和由通项公式（位置与项的关系）定义的数列。这两种类型均在 GCSE 进阶数学中出现。
| Sequence / 数列 | nth term / 通项 uₙ | Sum of n terms / 前 n 项和 Sₙ |
|---|---|---|
| Arithmetic / 等差 | a + (n−1)d | n/2 [2a + (n−1)d] |
| Geometric / 等比 | arⁿ⁻¹ | a(1 − rⁿ)/(1 − r) |
8. Inequalities | 不等式
Solving linear inequalities follows similar steps to equations, but multiplying or dividing by a negative number reverses the inequality sign. The solution set can be represented on a number line or in set notation.
解一元一次不等式的步骤与方程类似，但当两边同乘或同除以一个负数时，不等号方向要改变。解集可用数轴或集合符号表示。
Quadratic inequalities, e.g., x² − 4x + 3 > 0, are solved by factorising, finding critical values, and using a sign chart or sketching the graph. The solution is typically in two separate intervals.
二次不等式，例如 x² − 4x + 3 > 0，可通过因式分解、求界点，然后利用符号表或画草图来求解。解通常为两个不相交的区间。
When an inequality involves algebraic fractions, find a common denominator, bring all terms to one side, and consider the sign of the combined fraction. Never multiply through by a denominator whose sign is unknown.
当不等式中含代数分式时，先通分，将所有项移到一边，再考虑综合分式的符号。切勿在分母符号未知的情况下两边同乘以分母。
Graphical inequalities: shade the region that satisfies a given inequality. For y > f(x), shade above the curve; for y < f(x), shade below. Dashed line for strict inequalities, solid for ≤ or ≥.
图像不等式：为满足给定不等式的区域涂上阴影。对于 y > f(x)，在曲线上方涂色；对于 y < f(x)，在曲线下方涂色。严格不等式用虚线，带等号的不等式用实线。
9. Functions | 函数
A function maps every element of the domain to exactly one element of the range. The notation f(x) = x² + 3 defines a function. f(x) is not the same as the equation y = x² + 3; the former emphasises the mapping.
函数将定义域中的每个元素恰好映射到值域中的一个元素。符号 f(x) = x² + 3 定义了一个函数。f(x) 与方程 y = x² + 3 并不完全相同，前者强调映射关系。
Composite functions: gf(x) means apply f first, then g. (gf)(x) = g(f(x)). The order matters; fg(x) is generally different. The domain of the composite must respect the domain of the inner function and the outer function.
复合函数：gf(x) 表示先作用于 f，再作用于 g。(gf)(x) = g(f(x))。顺序很重要；fg(x) 通常与之不同。复合函数的定义域必须同时满足内层函数和外层函数的定义域。
Inverse functions: f⁻¹(x) reverses the effect of f(x). To find it, write y = f(x), swap x and y, then solve for y. The domain of f⁻¹ is the range of f. The graphs of f and f⁻¹ are reflections in the line y = x.
反函数：f⁻¹(x) 撤销 f(x) 的作用。求反函数时，令 y = f(x)，交换 x 和 y，再解出 y。f⁻¹ 的定义域是 f 的值域。f 与 f⁻¹ 的图像关于直线 y = x 对称。
Transformations of functions: y = f(x) + a translates vertically; y = f(x + a) translates horizontally (opposite direction); y = af(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a. Reflections: y = −f(x) reflects in x-axis, y = f(−x) reflects in y-axis.
函数变换：y = f(x) + a 纵向平移；y = f(x + a) 横向平移（方向相反）；y = af(x) 纵向拉伸因子 a；y = f(ax) 横向拉伸因子 1/a。反射：y = −f(x) 关于 x 轴对称，y = f(−x) 关于 y 轴对称。
10. Exponentials and Logarithms | 指数与对数
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(1/n) = √[n](a). The special base e ≈ 2.718 appears in the exponential function eˣ, which is its own derivative.
指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ / aⁿ = aᵐ⁻ⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a^(1/n) = ⁿ√a。特殊底数 e ≈ 2.718 出现在指数函数 eˣ 中，该函数的导数为其自身。
Logarithms are the inverses of exponentials: if aˣ = b, then logₐ b = x. Key rules: logₐ (xy) = logₐ x + logₐ y; logₐ (x/y) = logₐ x − logₐ y; logₐ (xⁿ) = n logₐ x. The natural log ln x has base e.
对数是指数的逆运算：若 aˣ = b，则 logₐ b = x。关键法则：logₐ (xy) = logₐ x + logₐ y；logₐ (x/y) = logₐ x − logₐ y；logₐ (xⁿ) = n logₐ x。自然对数 ln x 的底数为 e。
Solving exponential equations: take logs of both sides. Example: 2ˣ = 10 → x ln 2 = ln 10 → x = ln 10 / ln 2. Logarithmic equations often require combining logs into a single logarithm.
解指数方程：两边取对数。例如：2ˣ = 10 → x ln 2 = ln 10 → x = ln 10 / ln 2。对数方程通常需要合并对数成为一个单一的对数表达式。
The function ln x is the inverse of eˣ; its derivative is 1/x, and ∫ 1/x dx = ln|x| + C. These appear frequently in calculus contexts within Further Maths.
函数 ln x 是 eˣ 的反函数；其导数为 1/x，∫ 1/x dx = ln|x| + C。这些内容在进阶数学的微积分中经常出现。
11. Binomial Expansion | 二项展开式
For a positive integer n, (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … + xⁿ. This expansion is finite. The general term is given by the binomial coefficient: nCᵣ xʳ.
对于正整数 n，(1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … + xⁿ。此展开式是有限的。一般项由二项式系数给出：nCᵣ xʳ。
When the expression is (a + b)ⁿ, factor out a: aⁿ(1 + (b/a))ⁿ, then use the binomial theorem. The expansion is valid for all real x when n is a positive integer.
当表达式为 (a + b)ⁿ 时，提取因子 a 得到 aⁿ(1 + (b/a))ⁿ，再使用二项式定理。当 n 为正整数时，展开式对所有实数 x 均成立。
Pascal’s triangle provides the coefficients for small n, but the formula ⁿCᵣ = n! / [r!(n−r)!] is essential for larger n or when a specific term is requested. Remember the symmetrical property: ⁿCᵣ = ⁿCₙ₋ᵣ.
帕斯卡三角形提供了较小 n 的系数，但对于较大的 n 或需要特定项时，必须使用公式 ⁿCᵣ = n! / [r!(n−r)!]。记住对称性：ⁿCᵣ = ⁿCₙ₋ᵣ。
(1 + x)ⁿ = Σ (ⁿCᵣ) xʳ, r = 0 to n
12. Miscellaneous Key Formulae | 其他关键公式
Area of a triangle: besides ½ × base × height, use ½ ab sin C or Heron’s formula (area = √[s(s−a)(s−b)(s−c)], where s = semi-perimeter). Heron is less common at GCSE but may be derived.
三角形面积：除 ½ × 底 × 高外，还可使用 ½ ab sin C 或海伦公式（面积 = √[s(s−a)(s−b)(s−c)]，其中 s 为半周长）。海伦公式在 GCSE 中较少见，但可推导。
Volume of a sphere: V = ⁴⁄₃ π r³. Surface area of a sphere: A = 4π r². Volume of a cone: V = ⅓ π r² h. Curved surface area of a cone: A = π r l, where l is slant height.
球的体积：V = ⁴⁄₃ π r³。球的表面积：A = 4π r²。圆锥的体积：V = ⅓ π r² h。圆锥的侧面积：A = π r l，其中 l 为斜高。
Trapezium rule (approximate area under curve): A ≈ ½ h [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b−a)/n. This numerical method is used when integration is difficult.
梯形法则（近似曲线下面积）：A ≈ ½ h [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b−a)/n。此数值方法在积分困难时使用。
Circle theorem reminders: angle at the centre is twice the angle at the circumference; angles in the same segment are equal; angle in a semicircle is 90°; tangents and radii are perpendicular.
圆定理提示：圆心角是圆周角的两倍；同弧上的圆周角相等；半圆内的圆周角是 90°；切线与半径垂直。
Trigonometric equations often need the general solution in degrees or radians. Remember to use the CAST diagram or the graphs of sin, cos, tan to find all solutions within a given interval.
三角方程常需用角度制或弧度制求通解。记住使用 CAST 图或 sin、cos、tan 的图像来找出给定区间内的所有解。
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