📚 IGCSE CIE Further Mathematics: Quick-Reference Formula & Theorem Handbook | IGCSE CIE 进阶数学：公式定理速查手册
This handbook compiles essential formulas and theorems for the Cambridge IGCSE Further Mathematics (0606) examination, serving as a rapid revision tool. Each entry is presented bilingually for clarity, covering algebra, functions, trigonometry, calculus, vectors, and combinatorics. Use it to reinforce your understanding and boost exam confidence.
本手册汇编了剑桥 IGCSE 进阶数学（0606）考试必备公式与定理，作为快速复习工具。每个条目都以双语呈现，方便理解，涵盖代数、函数、三角、微积分、向量和排列组合等内容。用它来巩固理解并增强考试信心。
1. Algebra & Polynomials | 代数与多项式
The quadratic formula solves ax² + bx + c = 0 and is given by x = [-b ± √b² - 4ac] / (2a).
二次方程求根公式：ax² + bx + c = 0 的解为 x = [-b ± √b² - 4ac] / (2a)。
es two distinct real roots, Δ = 0 gives one repeated real root, Δ < 0 gives no real roots.
判别式 Δ = b² - 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
Completing the square rewrites a quadratic as a(x + h)² + k, where the vertex is at (-h, k).
配方法将二次式写成 a(x + h)² + k 的形式，其顶点坐标为 (-h, k)。
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a1/n = ⁿ√a, am/n = (ⁿ√a)ᵐ.
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a1/n = ⁿ√a，am/n = (ⁿ√a)ᵐ。
tive integer n: (a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + ... + ⁿCₙ bⁿ, where ⁿCᵣ = n! / [r!(n - r)!].
二项式定理（n为正整数）：(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + ... + ⁿCₙ bⁿ，其中 ⁿCᵣ = n! / [r!(n - r)!]。
For small x, (1 + x)ⁿ ≈ 1 + nx + [n(n-1)/2] x², and the expansion is valid when |x| < 1.
对于小量 x，(1 + x)ⁿ ≈ 1 + nx + [n(n-1)/2] x²，该展开在 |x| < 1 时有效。
² - b² = (a + b)(a - b); a³ + b³ = (a + b)(a² - ab + b²); a³ - b³ = (a - b)(a² + ab + b²).
因式分解恒等式：a² - b² = (a + b)(a - b)；a³ + b³ = (a + b)(a² - ab + b²)；a³ - b³ = (a - b)(a² + ab + b²)。
Partial fractions: for distinct linear factors, e.g. (5x+1)/[(x+2)(x-3)] ≡ A/(x+2) + B/(x-3). For repeated linear factors, e.g. (2x+3)/(x+1)² ≡ +1)². When the numerator’s degree ≥ denominator’s, perform polynomial long division first.
部分分式：对于不同线性因子，如 (5x+1)/[(x+2)(x-3)] ≡ A/(x+2) + B/(x-3)。对于重复线性因子，如 (2x+3)/(x+1)² ≡ A/(x+1) + B/(x+1)²。当分子次数 ≥ 分母次数时，先进行多项式长除法。
2. Equations & Inequalities | 方程与不等式
Sum of roots of ax² + bx + c = 0: α + β = -b/a; product of roots: αβ = c/a.
二次方程 ax² + bx + c = 0 的根之和：α + β = -b/a；根之积：αβ = c/a。
Common methods to solve quadratic equations: factorization, completing the square, and the quadratic formula.
解二次方程的常用方法：因式分解、配方法、求根公式。
isolate x; remember to flip the inequality sign when multiplying or dividing by a negative number.
对于线性不等式如 3x - 5 ≥ 7，将含 x 项移到一边；注意当乘以或除以负数时，不等号方向要改变。
Quadratic inequalities: solve the associated equation, then test intervals or use a sign chart. Example: x² - 4x + 3 < 0 gives solution 1 < x < 3.
二次不等式：先解对应方程，再检验区间或用符号表。例如 x² - 4x + 3 < 0 的解为 1 < x < 3。
Absolute value equations: |f(x)| = k implies f(x) = k or f(x) = -k, provided k ≥ 0. For absolute inequalities, |f(x)| < k means -k < f(x) < k; |f(x)| > k means f(x) < -k or f(x) > k.
绝对值方程：|f(x)| = k 等价于 f(x) = k 或 f(x) = -k，前提是 k ≥ 0。绝对值不等式 |f(x)| < k 转化成 -k < f(x) < k；|f(x)| > k 转化成 f(x) < -k 或 f(x) > k。
Simultaneous equations: one linear and one quadratic. Solve by substituting the linear expression into the quadratic equation; watch for extraneous solutions.
联立方程组（一个线性一个二次）：将线性表达式代入二次方程求解；注意排除增根。
3. Functions | 函数
A function f maps each element x from its domain to a unique element f(x) in its range. Domain: set of allowed inputs; Range: set of all possible outputs.
函数 f 将定义域中的每个元素 x 唯一映射到值域中的元素 f(x)。定义域：允许输入值的集合；值域：所有可能输出的集合。
Composite function: (g ◦ f)(x) = g(f(x)). Apply f first, then g.
复合函数：(g ◦ f)(x) = g(f(x))，先作用 f，再作用 g。
d it, write y = f(x), swap x and y, then solve for y. The domain of f⁻¹ is the range of f.
反函数 f⁻¹(x) 逆转 f 的作用。求法：令 y = f(x)，交换
Published by TutorHao | IGCSE 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图