GCSE CAIE Additional Mathematics: Formula & Theorem Quick Reference Handbook | GCSE CAIE 进阶数学：公式定理速查手册

📚 GCSE CAIE Additional Mathematics: Formula & Theorem Quick Reference Handbook | GCSE CAIE 进阶数学：公式定理速查手册

This quick reference handbook compiles all essential formulae and theorems from the CAIE IGCSE Additional Mathematics (0606) syllabus, organised for rapid revision. Use it to reinforce your memory before exams and to check key concepts while solving problems.

本速查手册汇编了 CAIE IGCSE 进阶数学（0606）课程大纲中所有核心公式与定理，按主题整理，便于快速复习。考前用它巩固记忆，解题时核对关键概念。

1. Algebra and Functions | 代数与函数

Function Notation: f : x ↦ f(x), where x belongs to the domain and f(x) belongs to the range.

函数记法：f : x ↦ f(x)，其中 x 属于定义域，f(x) 属于值域。

Composite Function: (f ∘ x)). The domain is the set of x in the domain of g for which g(x) lies in the domain of f.

复合函数：(f ∘ g)(x) = f(g(x))。其定义域为 g 的定义域中使得 g(x) 属于 f 的定义域的所有 x。

) and solving for y. f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. The domain of f⁻¹ is the range of f.

反函数：通过互换 y = f(x) 中的 x 和 y 并解出 y 得到 f⁻¹(x)。满足 f(f⁻¹(x)) = x 且 f⁻¹(f(x)) = x。f⁻¹ 的定义域是 f 的值域。

Absolute Value Function: |x| = x for x ≥ 0, and |x| = -x for x < 0. Its graph is V‑shaped.

绝对值函数：|x| = x（当 x ≥ 0），|x| = -x（当 x < 0）。其图像为 V 形。

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2. Quadratic Equations | 二次方程

Standard Form: ax² + bx + c = 0, with a ≠ 0.

标准式：ax² + bx + c = 0，其中 a ≠ 0。

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

判别式：Δ = b² − 4ac。Δ > 0 ⇒ 两个不等实根；Δ = 0 ⇒ 一个重根；Δ < 0 ⇒ 无实根。

Quadratic Formula: x = [ −b ± √b² − 4ac ] / (2a).

二次公式：x = [ −b ± √b² − 4ac ] / (2a)。

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

根的和与积：若根为 α 和 β，则 α + β = −b/a 且 αβ = c/a。

pleting the Square: ax² + bx + c = a(x − h)² + k, where h = −b/(2a) and k = c − (b²/(4a)).

完全平方：ax² + bx + c = a(x − h)² + k，其中 h = −b/(2a)，k = c − (b²/(4a))。

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3. Equations, Inequalities and Graphs | 方程、不等式与图像

Absolute Equation: |ax + b| = c leads to ax + b = c or ax + b = −c.

绝对值方程：|ax + b| = c 化为 ax + b = c 或 ax + b = −c。

Linear Inequality: When multiplying or dividing by a negative number, reverse the inequality sign.

线性不等式：乘以或除以负数时，不等号方向反转。

Quadratic Inequality: Solve ax² + bx + c > 0 by sketching the parabola. The solution is the x‑values where the graph is above the x‑axis.

二次不等式：画出抛物线草图来解 ax² + bx + c > 0。解集为图像在 x 轴上方的 x 值。

Cubic Graph: y = ax³ + bx² + cx + d. Find x‑intercepts by solving f(x) = 0 using factorisation.

三次图像：y = ax³ + bx² + cx + d。通过因式分解解 f(x) = 0 求 x 轴截距。

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4. Indices and Surds | 指数与根式

Laws of Indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.

指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，a⁰ = 1（a ≠ 0），a⁻ⁿ = 1/aⁿ。

Fractional Indices: a^m/n = ⁿ√aᵐ = (ⁿ√a)ᵐ.

分指数：a^m/n = ⁿ√aᵐ = (ⁿ√a)ᵐ。

Surd Simplification: √ab = √a × √b, √a/b = √a / √b. Rationalise denominators: 1/√a → √a/a; 1/(a+√b) → (a−√b)/(a²−b).

根式化简：√ab = √a × √b，√a/b = √a / √b。分母有理化：1/√a → √a/a；1/(a+√b) → (a−√b)/(a²−b)。

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5. Polynomials and Remainder Theorem | 多项式与余式定理

Remainder Theorem: When polynomial P(x) is divided by (x − a), the remainder is P(a).

余式定理：多项式 P(x) 除以 (x − a) 的余式为 P(a)。

Factor Theorem: If P(a) = 0, then (x − a) is a factor of P(x).

因式定理：若 P(a) = 0，则 (x − a) 是 P(x) 的因式。

= 0 by testing factors of the constant term, then use long division or synthetic division.

三次因式分解：通过检验常数项因数解 P(x) = 0，然后使用长除法或综合除法。

Polynomial Identity: Two polynomials are identical if coefficients of corresponding powers of x are equal.

多项式恒等：若两个多项式对应 x 次幂的系数相等，则它们恒等。

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6. Simultaneous Equations | 联立方程

Linear Systems: Solve by substitution or elimination. Exactly one solution exists unless lines are parallel or identical.

线性方程组：用代入法或消元法求解。除非两直线平行或重合，否则有唯一解。

One Linear & One Quadratic: Substitute the linear expression into the quadratic to form a quadratic equation in one variable; solve and back‑substitute.

一次与二次联立：将一次表达式代入二次，得到一元二次方程；求解并代回。

Geometric Interpretation: The solutions correspond to the intersection points of the graphs (line and parabola, line and circle, etc.).

几何解释：解对应图像的交点（如直线与抛物线、直线与圆等）。

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7. Logarithmic and Exponential Functions | 对数与指数函数

Definition: If aˣ = y, then x = logₐ y, where a > 0, a ≠ 1.

定义：若 aˣ = y，则 x = logₐ y，其中 a > 0 且 a ≠ 1。

Log Laws: logₐ (xy) = logₐ x + logₐ y; logₐ (x/y) = logₐ x − logₐ y; logₐ xⁿ = n logₐ x.

对数法则：logₐ (xy) = logₐ x + logₐ y；logₐ (x/y) = logₐ x − logₐ y；logₐ xⁿ = n logₐ x。

Change of Base: logₐ b = logₓ b / logₓ a, for any base c > 0, c ≠ 1.

换底公式：log

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