Year 9 CIE Further Pure Mathematics: Quick Reference of Formulae and Theorems | Year 9 CIE 进阶数学：公式定理速查手册

📚 Year 9 CIE Further Pure Mathematics: Quick Reference of Formulae and Theorems | Year 9 CIE 进阶数学：公式定理速查手册

This handbook brings together the essential formulae, identities and theorems that Year 9 CIE Further Pure Mathematics students need at their fingertips. It is organised as a quick revision reference covering algebra, quadratic equations, inequalities, indices and logarithms, functions, coordinate geometry, sequences and series, trigonometry, calculus, vectors and matrices. Use it to check your working, commit standard results to memory, and prepare efficiently for assessments.

本手册汇集了 Year 9 CIE 进阶纯数课程必备的公式、恒等式与定理，是一份随手可查的快速复习资料，涵盖代数、二次方程、不等式、指数与对数、函数、坐标几何、数列与级数、三角学、微积分、向量与矩阵等核心内容。可用它核对演算步骤、记忆标准结论，并高效备考。

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1. Algebraic Identities and Expansions | 代数恒等式与展开

Square of a sum: (a + b)² = a² + 2ab + b².

和平方公式： (a + b)² = a² + 2ab + b²。

Square of a difference: (a − b)² = a² − 2ab + b². Note that the middle term is negative but the last term is always positive.

差平方公式： (a − b)² = a² − 2ab + b²。注意中间项取负号，但末项恒为正。

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

平方差公式： a² − b² = (a − b)(a + b)。

Cube of a sum: (a + b)³ = a³ + 3a²b + 3ab² + b³.

和立方公式： (a + b)³ = a³ + 3a²b + 3ab² + b³。

Cube of a difference: (a − b)³ = a³ − 3a²b + 3ab² − b³. The signs alternate: +, −, +, −.

差立方公式： (a − b)³ = a³ − 3a²b + 3ab² − b³。符号交替出现：+、−、+、−。

Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²).

立方和公式： a³ + b³ = (a + b)(a² − ab + b²)。

Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²).

立方差公式： a³ − b³ = (a − b)(a² + ab + b²)。

Useful factorisation pattern: x² + (p + q)x + pq = (x + p)(x + q). This is the fastest route when the coefficient of x² is 1.

常用因式分解模式： x² + (p + q)x + pq = (x + p)(x + q)。当二次项系数为 1 时，这是最快的分解路径。

(a + b)ⁿ : coefficients follow Pascal's triangle

Pascal's triangle rows give the coefficients 1; 1, 1; 1, 2, 1; 1, 3, 3, 1; 1, 4, 6, 4, 1 and so on, which is exactly why the cube expansions above contain 1, 3, 3, 1.

杨辉三角（帕斯卡三角）给出系数 1；1, 1；1, 2, 1；1, 3, 3, 1；1, 4, 6, 4, 1 等，这正是上面立方展开中出现 1, 3, 3, 1 的原因。

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2. Quadratic Equations | 二次方程

Standard form: ax² + bx + c = 0, where a ≠ 0. Always rearrange into this form before choosing a method.

标准形式： ax² + bx + c = 0，其中 a ≠ 0。选择解法之前务必先化为该形式。

Discriminant: Δ = b² − 4ac. If Δ > 0, two distinct real roots; Δ = 0, one repeated real root; Δ < 0, no real roots.

判别式： Δ = b² − 4ac。若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重实根；若 Δ < 0，无实根。

Quadratic formula:

x = (−b ± √b² − 4ac) ÷ (2a)

求根公式： x = (−b ± √b² − 4ac) ÷ (2a)。使用前请确认根号内为 b² − 4ac，分母为 2a（而非 2 或 a）。

Sum and product of roots: If α and β are the roots, then α + β = −b/a and αβ = c/a.

根与系数的关系： 若 α 与 β 为方程的根，则 α + β = −b/a，αβ = c/a。

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

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

Vertex from completed square: for y = a(x − h)² + k, the turning point is (h, k). When a > 0 it is a minimum; when a < 0 it is a maximum.

由配方读出顶点： 对于 y = a(x − h)² + k，顶点为 (h, k)。当 a > 0 时为最小值点；当 a < 0 时为最大值点。

Choosing a method: factorise when the roots are rational and obvious; complete the square when you need the vertex or an exact surd form; use the formula when factorising stalls.

选择解法： 根为明显有理数时用因式分解；需要顶点或精确根式形式时用配方法；分解受阻时用求根公式。

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3. Inequalities | 不等式

Basic properties: If a < b then a + c < b + c. If c > 0, then ac < bc; if c < 0, then ac > bc.

基本性质： 若 a < b，则 a + c < b + c。若 c > 0，则 ac < bc；若 c < 0，则 ac > bc。

The golden rule: multiplying or dividing an inequality by a negative number reverses the direction of the sign. This is the single most common source of lost marks.

黄金法则： 不等式两边乘以或除以负数时，不等号方向必须反转。这是最常见的失分点。

Quadratic inequalities: solve ax² + bx + c = 0 first to find the critical values, then sketch the parabola. For a > 0, the expression is negative between the roots and positive outside them.

二次不等式： 先解 ax² + bx + c = 0 得到临界值，再画抛物线草图。当 a > 0 时，表达式在两根之间为负、在两根之外为正。

Worked pattern: (x − 2)(x + 3) > 0 gives x < −3 or x > 2, while (x − 2)(x + 3) < 0 gives −3 < x < 2.

典型结果： (x − 2)(x + 3) > 0 得 x < −3 或 x > 2；而 (x − 2)(x + 3) < 0 得 −3 < x < 2。

Rational inequalities: never multiply across by an expression whose sign is unknown. Move everything to one side, combine over a single denominator, and use a sign diagram with the critical values marked.

分式不等式： 切勿乘以符号未知的整式。应将所有项移到一侧，通分合并为一个分式，再画符号图并标出临界值。

Modulus inequalities: |x| < k (with k > 0) means −k < x < k, while |x| > k means x < −k or x > k. For |x − a| < k, the interval of solutions is (a − k, a + k).

绝对值不等式： 当 k > 0 时，|x| < k 表示 −k < x < k；|x| > k 表示 x < −k 或 x > k。对于 |x − a| < k，解集区间为 (a − k, a + k)。

Answer presentation: on a number line, and check whether the endpoints are included (≤, ≥) or excluded (<, >).

答案呈现： 解集可写成不等式、区间或数轴形式，并注意端点是否包含（≤、≥ 包含；<、> 不包含）。

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4. Indices, Surds and Logarithms | 指数、根式与对数

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

指数运算法则： aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。

Fractional indices: a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√aᵐ = (ⁿ√a)ᵐ. The denominator of the power is the root; the numerator is the power.

分数指数： a^(1/n) = ⁿ√a，a^(m/n) = ⁿ√aᵐ = (ⁿ√a)ᵐ。指数中的分母代表开方，分子代表乘方。

Surd rules: √a × √b = √ab, √a ÷ √b = √a/b, and √a²b = a√b for a > 0. Always simplify surds before adding them.

根式法则： √a × √b = √ab，√a ÷ √b = √a/b，当 a > 0 时 √a²b = a√b。相加之前务必先化简根式。

Rationalising denominators: multiply by the conjugate. For 1/(a + √b) use (a − √b), since (a + √b)(a − √b) = a² − b, which is rational.

分母有理化： 乘以共轭式。对于 1/(a + √b)，乘以 (a − √b)，因为 (a + √b)(a − √b) = a² − b，结果为有理数。

Definition of a logarithm: if aˣ = N then logₐN = x, where a > 0 and a ≠ 1. A logarithm is simply an exponent.

对数的定义： 若 aˣ = N，则 logₐN = x，其中 a > 0 且 a ≠ 1。对数本质上就是一个指数。

Laws of logarithms: log(MN) = logM + logN, log(M/N) = logM − logN, log(Mᵏ) = k logM, and logₐ1 = 0, logₐa

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