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Year 9 Edexcel Further Maths: Formula and Theorem Quick Reference | Year 9 Edexcel 进阶数学：公式定理速查手册

2026-07-18 07:04 · 本文部分内容由AI辅助创作

📚 Year 9 Edexcel Further Maths: Formula and Theorem Quick Reference | Year 9 Edexcel 进阶数学：公式定理速查手册

This quick reference guide compiles the essential formulas and theorems covered in the Year 9 Edexcel Further Maths curriculum. It is designed to help you revise key concepts efficiently, with each formula presented in a clear bilingual format, so that you can move between the English terminology used in examinations and the Chinese explanation used for understanding.

本速查手册汇总了 Year 9 Edexcel 进阶数学课程中的核心公式与定理，旨在帮助你高效复习关键概念，每条公式均以清晰的中英双语呈现，方便你在考试所用的英文术语与帮助理解的中文解释之间自由切换。

1. Algebra Fundamentals | 代数基础

Expanding two brackets follows the distributive law: multiply every term in the first bracket by every term in the second bracket. For (a + b)(c + d) this produces four products, and like terms must always be collected afterwards.

展开两个括号遵循分配律：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) 会产生四项乘积，展开之后务必合并同类项。

(a + b)(c + d) = ac + ad + bc + bd

The perfect square expansions appear in almost every later topic, so they must be memorised. Note that the middle term is twice the product of the two terms inside the bracket.

完全平方展开式几乎出现在之后所有主题中，因此必须牢记。注意中间项恰是括号内两项乘积的两倍。

(a + b)² = a² + 2ab + b²    (a − b)² = a² − 2ab + b²

hidden difference of squares such as x⁴ − 16 = (x² − 4)(x² + 4) = (x − 2)(x + 2)(x² + 4).

平方差公式可分解为和与差的乘积。务必先检查是否有公因子，并留意隐藏的平方差形式，例如 x⁴ − 16 = (x² − 4)(x² + 4) = (x − 2)(x + 2)(x² + 4)。

a² − b² = (a − b)(a + b)

Factorising quadratics of the form x² + bx + c: find two numbers that multiply to give c and add to give b, then write (x + p)(x + q).

形如 x² + bx + c 的二次式因式分解：找到两个数，它们的积等于 c、和等于 b，然后写成 (x + p)(x + q)。

Completing the square rewrites any quadratic so that its turning point can be read off directly, which is essential when sketching graphs without a calculator.

配方法把任意二次式改写为可以直接读出顶点坐标的形式，这在不用计算器画图像时极为关键。

x² + bx + c = (x + b/2)² + c − b²/4

2. Indices, Surds and Standard Form | 指数、根式与科学计数法

The index laws are the backbone of algebraic manipulation and appear in every paper. Learn them as rules about adding, subtracting and multiplying powers of the same base.

指数法则是代数变形的骨架，几乎每份试卷都会出现。要把它们记成关于同底数幂的加减与相乘规则。

aᵐ × aⁿ = aᵐ⁺ⁿ    aᵐ ÷ aⁿ = aᵐ⁻ⁿ    (aᵐ)ⁿ = aᵐⁿ

a⁰ = 1    a⁻ⁿ = 1/aⁿ    a1/n = ⁿ√a    am/n = ⁿ√aᵐ

A surd is an irrational root such as √2 or √50. Simplify by extracting the largest square factor, and rationalise a denominator by multiplying top and bottom by the conjugate.

根式是指像 √2 或 √50 这样的无理根。化简时提取最大的平方因子；分母有理化时上下同乘共轭式。

It is required for very large or very small quantities and for significant-figure questions.

科学计数法把数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。它用于表示极大或极小的量，也常用于有效数字相关的问题。

3. Linear Equations, Inequalities and Simultaneous Equations | 一次方程、不等式与联立方程

Solving a linear equation means isolating the unknown by applying inverse operations in reverse order. Always check the answer by substitution.

解一次方程就是用逆运算按相反顺序把未知数孤立出来。解完一定要代回原式检验。

ax + b = c   ⟹   x = (c − b)/a   (a ≠ 0)

Inequalities behave exactly like equations, with one critical exception: multiplying or dividing by a negative number reverses the direction of the inequality sign.

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不等式的运算与方程完全相同，只有一个关键例外：乘以或除以负数时，不等号方向必须反转。

Simultaneous equations in two unknowns are solved either by elimination (matching coefficients and subtracting) or by substitution (rearranging one equation and inserting it into the other).

二元联立方程可用消元法（配平系数后相减）或代入法（先变形一个方程再代入另一个）求解。

and a₂x + b₂y = c₂, compare the ratios of coefficients to decide how many solutions exist.

对于方程组 a₁x + b₁y = c₁ 与 a₂x + b₂y = c₂，可通过比较系数之比判断解的个数。

a₁/a₂ ≠ b₁/b₂ One unique solution. 唯一解。
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ No solution (parallel lines). 无解（两直线平行）。
a₁/a₂ = b₁/b₂ = c₁/c₂ Infinitely many solutions (same line). 无穷多解（同一直线）。

4. Sequences and Series | 数列与级数

An arithmetic sequence has a constant common difference d, so each term is obtained by adding d to the previous term. The nth term formula lets you jump straight to any position without listing every term.

等差数列有恒定的公差 d，每一项都是前一项加 d 得到。通项公式让你不必逐项列举就能直接求出任意位置的项。

aₙ = a₁ + (n − 1)d

The sum of the first n terms of an arithmetic series can be found from the first term and the last term, which is often quicker than using a₁ and d.

等差数列前 n 项之和可以用首项与末项求出，通常比用 a₁ 和 d 更快。

Sₙ = n/2 × (2a₁ + (n − 1)d) = n/2 × (a₁ + aₙ)

A geometric sequence multiplies by a constant ratio r each time. If |r| < 1 the terms shrink towards zero; if |r| > 1 they grow rapidly.

等比数列每次乘以恒定公比 r。若 |r| < 1，各项趋于 0；若 |r| > 1，各项迅速增大。

aₙ = a₁ rⁿ⁻¹    Sₙ = a₁(1 − rⁿ)/(1 − r)   (r ≠ 1)

sing on sight: square numbers n², cube numbers n³, and triangular numbers Tₙ = n(n + 1)/2.

有些特殊数列要一眼认出：平方数 n²、立方数 n³，以及三角数 Tₙ = n(n + 1)/2。

5. Coordinate Geometry of Straight Lines | 直线坐标几何

Coordinate geometry connects algebra and shape. The gradient measures steepness, and a negative gradient means the line falls from left to right.

坐标几何把代数与图形连接起来。斜率衡量陡峭程度，斜率为负表示直线从左向右下降。

m = (y₂ − y₁)/(x₂ − x₁)

The distance formula is simply Pythagoras applied to the horizontal and vertical gaps between two points, and the midpoint formula averages the coordinates.

距离公式其实就是对两点间的水平差与竖直差应用毕达哥拉斯定理，中点公式则取坐标的平均值。

d = √((x₂ − x₁)² + (y₂ − y₁)²)    M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

The equation of a straight line can be written in gradient-intercept form or, when a point and a gradient are known, in point-gradient form.

直线方程可写成斜截式；当已知一点与斜率时，也可用点斜式。

y = mx + c    y − y₁ = m(x − x₁)

Parallel lines share the same gradient, while perpendicular lines have gradients whose product is −1.

平行直线斜率相同，而垂直直线的斜率之积为 −1。

m₁ = m₂ (parallel 平行)    m₁ × m₂ = −1 (perpendicular 垂直)