CIE Year 10 Further Mathematics: Formula & Theorem Quick Reference | CIE Year 10 进阶数学：公式定理速查手册

📚 CIE Year 10 Further Mathematics: Formula & Theorem Quick Reference | CIE Year 10 进阶数学：公式定理速查手册

This quick-reference handbook compiles all essential formulas and theorems required for the CIE Year 10 Further Mathematics (Additional Mathematics) course. It covers algebra, functions, trigonometry, calculus, vectors, and more, serving as a concise revision tool for both coursework and examinations.

本速查手册汇编了 CIE Year 10 进阶数学（附加数学）课程所需的所有核心公式和定理，涵盖代数、函数、三角学、微积分、向量等内容，是课堂学习和备考复习的高效参考工具。

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1. Quadratic Functions & Equations | 二次函数与方程

A quadratic function has the general form f(x) = ax² + bx + c where a, b, c are real constants and a ≠ 0.

二次函数的一般式为 f(x) = ax² + bx + c，其中 a、b、c 为实数且 a ≠ 0。

Its graph is a parabola. When a > 0 the parabola opens upward (minimum point); when a < 0 it opens downward (maximum point).

其图像为抛物线。当 a > 0 时开口朝上（有最小值）；当 a < 0 时开口朝下（有最大值）。

The discriminant is given by Δ = b² − 4ac. It determines the nature of the roots of the quadratic equation ax² + bx + c = 0.

判别式为 Δ = b² − 4ac，它决定了二次方程 ax² + bx + c = 0 根的性质。

If Δ > 0, the equation has two distinct real roots. If Δ = 0, it has one repeated real root (a double root). If Δ < 0, it has no real roots (two complex conjugate roots).

若 Δ > 0，方程有两个不等实根；若 Δ = 0，有一个重实根；若 Δ < 0，无实根（一对共轭复根）。

The roots can be found using the quadratic formula: x = [−b ± √(b² − 4ac)] / (2a).

求根可使用 二次公式：x = [−b ± √(b² − 4ac)] / (2a)。

Turning point (vertex) coordinates: (−b/(2a) , f(−b/(2a))). By completing the square, f(x) = a(x − h)² + k gives the vertex (h, k).

顶点坐标：(−b/(2a) , f(−b/(2a)))。通过配方法得到 f(x) = a(x − h)² + k，顶点即为 (h, k)。

Sum of roots = −b/a, product of roots = c/a. For a quadratic inequality such as ax² + bx + c > 0, solve the corresponding equation and test intervals.

根的和 = −b/a，根的积 = c/a。对于二次不等式如 ax² + bx + c > 0，可先解对应方程，再通过区间检验求解集。

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2. Polynomials & Factor Theorem | 多项式与因式定理

For a polynomial P(x), the Remainder Theorem states that when P(x) is divided by (x − a), the remainder is P(a).

对于多项式 P(x)，余数定理指出：当除以 (x − a) 时，余数为 P(a)。

The Factor Theorem is a special case: (x − a) is a factor of P(x) if and only if P(a) = 0.

因式定理为特例：(x − a) 是 P(x) 的因式当且仅当 P(a) = 0。

Polynomial division can be performed by long division or by equating coefficients. If the divisor is quadratic, the remainder is of the form px + q.

多项式除法可使用长除法或比较系数法进行。若除式为二次式，则余式形如 px + q。

To factorise a cubic expression, find one factor using trial values (often ±1, ±2, ±1/2, etc.) and then divide to obtain a quadratic factor.

对三次多项式因式分解时，先通过试根（常取 ±1、±2、±½ 等）找到一个因式，再通过除法得到二次因式。

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3. Indices, Surds & Logarithms | 指数、根式与对数

Laws of indices for a > 0: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^(1/n) = ⁿ√a.

指数定律（a > 0）：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a^(1/n) = ⁿ√a。

Surd rules: √(ab) = √a × √b, √(a/b) = √a / √b (b ≠ 0). To rationalise a denominator such as 1/(√a + b), multiply numerator and denominator by the conjugate (√a − b).

根式运算：√(ab) = √a × √b，√(a/b) = √a / √b（b ≠ 0）。要将分母有理化，如 1/(√a + b)，分子分母同乘共轭根式 (√a − b)。

Definition of logarithm: if aˣ = b, then x = logₐb (a > 0, a ≠ 1, b > 0). The natural logarithm ln x has base e.

对数定义：若 aˣ = b，则 x = logₐb（a > 0，a ≠ 1，b > 0）。自然对数 ln x 以 e 为底。

Logarithm laws: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xⁿ) = n logₐx.

对数运算法则：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，logₐ(xⁿ) = n logₐx。

Change of base formula: logₐb = (log_c b) / (log_c a), often used with base 10 or e.

换底公式：logₐb = (log_c b) / (log_c a)，常取底数 10 或 e。

Solving exponential equations often requires taking logarithms on both sides; logarithmic equations may need to be combined into a single log.

解指数方程时常两边取对数；解对数方程时需合并为单一对数再化简。

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4. Coordinate Geometry | 坐标几何

For two points A(x₁, y₁) and B(x₂, y₂): midpoint = ((x₁+x₂)/2 , (y₁+y₂)/2) and distance AB = √[(x₂ − x₁)² + (y₂ − y₁)²].

对于两点 A(x₁, y₁) 和 B(x₂, y₂)：中点 = ((x₁+x₂)/2 , (y₁+y₂)/2)，距离 AB = √[(x₂ − x₁)² + (y₂ − y₁)²]。

The gradient (slope) of AB is m = (y₂ − y₁) / (x₂ − x₁).

AB 的斜率（坡度）为 m = (y₂ − y₁) / (x₂ − x₁)。

Forms of a straight line: y = mx + c (slope-intercept), y − y₁ = m(x − x₁) (point-slope), and ax + by + c = 0 (general form).

直线的几种形式：y = mx + c（斜截式），y − y₁ = m(x − x₁)（点斜式），ax + by + c = 0（一般式）。

Parallel lines have equal gradients: m₁ = m₂. Perpendicular lines satisfy m₁ × m₂ = −1 (provided neither is vertical).

平行直线的斜率相等：m₁ = m₂。垂直直线的斜率之积为 −1：m₁ × m₂ = −1（不含垂直于坐标轴的情形）。

The equation of a circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

圆心 (h, k)、半径 r 的圆的方程为 (x − h)² + (y − k)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0

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