📚 Year 9 OCR Further Maths: Formula & Theorem Quick Reference | 九年级OCR进阶数学：公式定理速查手册
This quick reference handbook compiles essential formulas, identities and theorems for the Year 9 OCR Further Maths course. Each entry is presented in a concise, exam-focused style to support rapid revision and classroom learning.
本速查手册汇编了九年级OCR进阶数学课程的核心公式、恒等式与定理，所有条目均以简明、贴近考试的风格呈现，便于快速复习与课堂巩固。
1. Algebraic Expressions and Identities | 代数表达式与恒等式
When simplifying algebraic expressions, like terms can be combined by adding or subtracting their coefficients. The distributive law a(b + c) = ab + ac is fundamental for expanding brackets.
化简代数表达式时，同类项可以通过系数相加减合并。分配律 a(b + c) = ab + ac 是展开括号的基础法则。
a(b + c) = ab + ac
Expanding a product of two binomials uses the FOIL method: (x + a)(x + b) = x² + (a+b)x + ab. Special products include perfect squares and difference of two squares.
两个二项式相乘时使用FOIL法则展开：(x + a)(x + b) = x² + (a+b)x + ab 。常见特殊展开包括完全平方公式和平方差公式。
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
a² − b² = (a + b)(a − b)
Factorising an expression is the reverse of expanding. Always look for a common factor first, then recognise any standard forms such as difference of squares or quadratic trinomials.
因式分解是展开的逆运算。应先提取公因子，再识别标准形式，如平方差或二次三项式。
2. Linear and Simultaneous Equations | 线性方程与联立方程
A linear equation can be solved by isolating the unknown. Use inverse operations: addition/subtraction then multiplication/division, keeping the equation balanced.
解一元一次方程可通过逆运算求解未知数：先加减后乘除，保持方程两边平衡。
The equation of a straight line is typically written as y = mx + c, where m is the gradient and c is the y-intercept. The general form ax + by + c = 0 is also used.
直线方程通常写作 y = mx + c ，其中 m 为斜率，c 为y轴截距。也使用一般式 ax + by + c = 0 。
m = (y₂ − y₁) / (x₂ − x₁)
Simultaneous linear equations in two variables can be solved by elimination or substitution. Multiplying equations to obtain equal coefficients often helps eliminate one variable.
含两个变量的联立一次方程可通过消元法或代入法求解。通常先将方程乘上适当倍数，使某个变量系数相等，再消元。
The solution to a pair of simultaneous equations represents the coordinates where their graphs intersect. Parallel lines have no solution, and coincident lines have infinitely many.
联立方程组的解代表两条直线图像的交点坐标。平行线无解，重合直线有无穷多解。
3. Quadratic Equations | 二次方程
A quadratic equation takes the form ax² + bx + c = 0, with a ≠ 0. It can be solved by factorising, completing the square, or using the quadratic formula.
二次方程的形式为 ax² + bx + c = 0 (a ≠ 0)。可通过因式分解、配方法或求根公式求解。
x = (−b ± √(b² − 4ac)) / (2a)
The expression b² − 4ac under the square root is called the discriminant (Δ). It determines the nature of the roots: two real distinct roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.
根号内的表达式 b² − 4ac 称为判别式 (Δ)。它决定根的性质：Δ > 0 时有两个相异实根，Δ = 0 时有一个重根，Δ < 0 时无实根。
Completing the square rewrites the quadratic as a(x + p)² + q = 0, revealing the vertex of its graph at (−p, q).
配方法将二次式写成 a(x + p)² + q = 0 的形式，其图像顶点为 (−p, q)。
Always look to factorise first: if the quadratic can be expressed as (px + q)(rx + s) = 0, then the solutions are x = −q/p and x = −s/r.
优先尝试因式分解：若二次式可化为 (px + q)(rx + s) = 0 ，则解为 x = −q/p 和 x = −s/r 。
4. Inequalities | 不等式
Inequalities use symbols >, <, ≥ and ≤. Solving them follows the same steps as linear equations, but multiplying or dividing by a negative number reverses the inequality sign.
不等式使用 >、<、≥ 和 ≤ 符号。其求解步骤与一次方程相同，但当乘以或除以负数时，不等号方向要反转。
Solutions can be represented on a number line: an open circle for strict inequalities, a closed circle for inclusive ones. Compound inequalities like a < x ≤ b describe an interval.
解集可用数轴表示：严格不等式用空心圆圈，包含等号的用实心圆点。复合不等式如 a < x ≤ b 描述一个区间。
To solve quadratic inequalities, first solve the corresponding equation, then test intervals between the critical values to determine where the inequality holds.
解二次不等式时，先求解对应的方程，然后在临界值划分的各区间内检验符号，确定不等式成立的范围。
5. Sequences and Series | 序列与级数
An arithmetic sequence increases or decreases by a constant common difference d. Its nth term is given by uₙ = a + (n − 1)d, where a is the first term.
等差数列以固定的公差 d 递增或递减。第 n 项的通项公式为 uₙ = a + (n − 1)d ，其中 a 为首项。
The sum of the first n terms of an arithmetic sequence is Sₙ = n/2 [2a + (n − 1)d] or, equivalently, Sₙ = n/2 (a + l) where l is the last term.
等差数列前 n 项的和为 Sₙ = n/2 [2a + (n − 1)d] ，或等价地 Sₙ = n/2 (a + l) ，其中 l 为末项。
Sₙ = n/2 (a + l)
Recognising whether a sequence is arithmetic often requires checking that the difference between consecutive terms remains constant. Patterns from diagrams can also generate sequences.
判断数列是否为等差数列通常需检查连续项之差是否恒定。根据图形规律也可生成数列。
6. Indices and Standard Form | 指数与标准型
The laws of indices simplify calculations with powers. They apply when bases are the same and the operation is multiplication, division or raising a power to another power.
指数法则用于简化幂运算，要求底数相同，可用于乘法、除法或幂的乘方。
| aᵐ × aⁿ = aᵐ⁺ⁿ | 同底数幂相乘，指数相加 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 同底数幂相除，指数相减 |
| (aᵐ)ⁿ = aᵐⁿ | 幂的乘方，指数相乘 |
| a⁰ = 1 (a ≠ 0) | 任何非零数的0次幂等于1 |
| a⁻ⁿ = 1 / aⁿ | 负指数表示倒数 |
Standard form writes a number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. It is used for very large or very small quantities.
标准型将一个数记为 a × 10ⁿ ，其中 1 ≤ a < 10 且 n 为整数。多用于很大或很小的数。
Calculations in standard form combine the decimal parts and the powers of ten separately, then adjust to keep the coefficient between 1 and 10.
在标准型下进行运算时，先将小数部分与10的幂次分别处理，再调整系数使其介于1与10之间。
7. Coordinate Geometry | 坐标几何
The distance between two points (x₁, y₁) and (x₂, y₂) is found using the distance formula, which is derived from Pythagoras’ theorem.
两点 (x₁, y₁) 和 (x₂, y₂) 间的距离由距离公式求出，该公式由勾股定理推导而来。
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
The midpoint M of a segment joining the two points has coordinates given by the average of the x-coordinates and the y-coordinates.
连接两点的线段中点 M 的坐标为 x 坐标与 y 坐标的平均值。
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The gradient m of the line through (x₁, y₁) and (x₂, y₂) is rise over run: m = (y₂ − y₁)/(x₂ − x₁).
过 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m 为纵坐标之差除以横坐标之差：m = (y₂ − y₁)/(x₂ − x₁) 。
Parallel lines have equal gradients. Two lines with gradients m₁ and m₂ are perpendicular if and only if m₁ × m₂ = −1.
平行线斜率相等。斜率分别为 m₁ 和 m₂ 的两条直线垂直当且仅当 m₁ × m₂ = −1 。
8. Pythagoras and Trigonometry | 勾股定理与三角学
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
在直角三角形中，斜边（直角对边）的平方等于两直角边的平方和。
a² + b² = c²
The converse is also true: if three sides satisfy a² + b² = c², the triangle is right-angled with c as the hypotenuse.
逆定理同样成立：若三边满足 a² + b² = c² ，则该三角形为直角三角形，c 为斜边。
Trigonometric ratios in a right-angled triangle relate an acute angle to the ratios of two sides. Use SOH CAH TOA to recall them:
直角三角形中的三角比将锐角与两边之比建立联系。使用 SOH CAH TOA 帮助记忆：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Exact values for special angles should be known for quick problem-solving:
需熟记特殊角的精确值以便快速解题：
| sin 30° = 1/2 | sin 45° = √2/2 | sin 60° = √
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