Year 11 Edexcel Further Maths: Quick-Reference Formula & Theorem Handbook | Edexcel 进阶数学公式定理速查手册

📚 Year 11 Edexcel Further Maths: Quick-Reference Formula & Theorem Handbook | Edexcel 进阶数学公式定理速查手册

This handbook provides a concise, examination-focused summary of all key formulas and theorems required for the Edexcel International GCSE Further Pure Mathematics specification (4PM1). It is designed for rapid revision and easy memorisation, with each entry presented in clear English and Chinese.

本手册紧扣Edexcel国际GCSE进阶纯数考试大纲，用中英双语简明归纳所有核心公式与定理，便于快速查阅与记忆。

1. Algebraic Manipulation & Surds | 代数运算与根式

Difference of two squares: a² - b² = (a + b)(a - b).

平方差公式：a² - b² = (a + b)(a - b)。

Perfect square expansions: (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b².

完全平方展开：(a + b)² = a² + 2ab + b²，(a - b)² = a² - 2ab + b²。

Rationalising denominators: for 1/√a, multiply numerator and denominator by √a; for 1/(a + √b), multiply by the conjugate (a - √b).

分母有理化：对1/√a，分子分母同乘√a；对1/(a + √b)，乘共轭根式(a - √b)。

Laws of indices: a^m × aⁿ = a^m+n, (a^m)ⁿ = a^mn, a^-n = 1/aⁿ, a^1/n = ⁿ√a.

指数律：a^m × aⁿ = a^m+n，(a^m)ⁿ = a^mn，a^-n = 1/aⁿ，a^1/n = ⁿ√a。

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2. Quadratic Equations & Functions | 二次方程与函数

The quadratic formula for ax² + bx + c = 0: x = [-b ± √b² - 4ac] / (2a).

二次方程求根公式：ax² + bx + c = 0 的解为 x = [-b ± √b² - 4ac] / (2a)。

0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.

判别式 Δ = b² - 4ac。若 Δ > 0，有两个不等实根；Δ = 0，一个重根；Δ < 0，无实根。

Sum and product of roots: if α and β are roots, then α + β = -b/a and αβ = c/a.

根与系数的关系：α + β = -b/a，αβ = c/a。

Completing the square: ax² + bx + c = a(x + b/(2a))² + (c - b²/(4a)).

配方法：ax² + bx + c = a(x + b/(2a))² + (c - b²/(4a))。

3. Coordinate Geometry: Straight Lines & Circles | 坐标几何：直线与圆

s: m = (y₂ - y₁)/(x₂ - x₁). Equation of a straight line: y - y₁ = m(x - x₁) or y = mx + c.

两点间斜率：m = (y₂ - y₁)/(x₂ - x₁)。直线方程：点斜式 y - y₁ = m(x - x₁) 或斜截式 y = mx + c。

Parallel lines have equal gradients; for perpendicular lines, m₁ × m₂ = -1.

平行线斜率相等；垂直线斜率乘积为 -1。

Midpoint of two points: ((x₁ + x₂)/2, (y₁ + y₂)/2). Distance: √[(x₂ - x₁)² + (y₂ - y₁)²].

中点坐标：( (x₁ + x₂)/2 , (y₁ + y₂)/2 )。两点距离：√[(x₂ - x₁)² + (y₂ - y₁)²]。

Equation of a circle with centre (a, b) and radius r: (x - a)² + (y - b)² = r².

圆的标准方程：(x - a)² + (y - b)² = r²，圆心 (a, b)，半径 r。

The general equation x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (-g, -

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