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

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

This comprehensive quick reference handbook brings together the essential formulas, theorems, and problem-solving tools you need for the Year 10 Edexcel Further Mathematics course. Use it to refresh your memory, spot connections between topics, and build confidence before assessments.

本综合速查手册汇集了 Year 10 Edexcel 进阶数学课程所需的核心公式、定理与解题工具。你可以用它来巩固记忆、发现知识点之间的联系，并在考前建立信心。

1. Algebraic Expressions & Surds | 代数表达式与根式

ᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ. Any non-zero base raised to the power zero equals 1.

指数定律：对 a > 0，aᵐ × aⁿ = aᵐ⁺ⁿ；(aᵐ)ⁿ = aᵐⁿ；a⁰ = 1；a⁻ⁿ = 1/aⁿ。任何非零底数的零次幂都等于 1。

Surds: √ab = √a √b and √a/b = √a / √b. Rationalising the denominator: 1/√a = √a/a ; for a binomial denominator, 1/(√a + √b) = (√a - √b)/(a - b).

根式：√ab = √a √b，√a/b = √a / √b。分母有理化：1/√a = √a/a；对于二项式分母，1/(√a + √b) = (√a - √b)/(a - b)。

Key expansions: (a ± b)² = a² ± 2ab + b² ; (a + b)(a - b) = a² - b².

核心乘法公式：(a ± b)² = a² ± 2ab + b²；(a + b)(a - b) = a² - b²。

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2. Quadratic Equations & Functions | 二次方程与函数

Standard form: ax² + bx + c = 0, a ≠ 0. The solutions are given by the quadratic formula:

标准形式：ax² + bx + c = 0，a ≠ 0。由求根公式给出解：

x = (-b ± √b² - 4ac) / (2a)

s: Δ > 0 → two distinct real roots; Δ = 0 → one repeated real root; Δ < 0 → no real roots.

判别式 Δ = b² - 4ac 决定根的性质：Δ > 0 → 两个不等实根；Δ = 0 → 一个重实根；Δ < 0 → 无实根。

of symmetry is x = -b/(2a). Completing the square rewrites the quadratic as a(x + p)² + q.

抛物线 y = ax² + bx + c 的顶点坐标：x 坐标 = -b/(2a)，y 坐标 = (4ac - b²)/(4a)。对称轴为 x = -b/(2a)。配方法可将二次式写成 a(x + p)² + q。

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3. Inequalities | 不等式

Linear inequalities are solved like linear equations, but multiply/divide by a negative number reverses the inequality sign. Solution sets can be expressed on a number line or using interval notation.

线性不等式的解法与线性方程类似，但乘以或除以负数时需反转不等号。解集可用数轴或区间符号表示。

ax² + bx + c > 0 by finding the critical values from ax² + bx + c = 0 and testing regions.

二次不等式通过画出抛物线草图，并确定图像在 x 轴上方或下方的区间来求解。解 ax² + bx + c > 0 时，先解方程 ax² + bx + c = 0 求出临界值，再检验各区间。

Absolute value inequalities: |ax + b| < c ⇔ -c < ax + b < c ; |ax + b| > c ⇔ ax + b < -c or ax + b > c.

绝对值不等式：|ax + b| < c ⇔ -c < ax + b < c ；|ax + b| > c ⇔ ax + b < -c 或 ax + b > c。

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4. Exponentials & Logarithms | 指数与对数

The logarithmic and exponential forms are equivalent: y = logₐx ⇔ aʸ = x, with a > 0, a ≠ 1. Common bases are 10 (log₁₀) and e (natural log, ln).

对数形式与指数形式等价：y = logₐx ⇔ aʸ = x，其中 a > 0, a ≠ 1。常用底数为 10（log₁₀）和 e（自然对数，记作 ln）。

ange of base: logₓb = logₐb / logₐa. Special values: logₐ1 = 0, logₐa = 1, logₐ(1/a) = -1.

对数运算律：logₐ(xy) = logₐx + logₐy ；logₐ(x/y) = logₐx - logₐy ；logₐ(xⁿ) = n logₐx。换底公式：logₓb = logₐb / logₐa。特殊值：logₐ1 = 0，logₐa = 1，logₐ(1/a) = -1。

ᵏ = m, then take logs both sides: k logₐa = logₐm. If using natural logs, k = ln m / ln a.

指数方程：将 aᵏ = m 分离后，两边取对数：k logₐa = logₐm。若使用自然对数，k = ln m / ln a。

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5. Sequences & Series | 序列与级数

d ; sum of first n terms Sₙ = n/2 [2a + (n - 1)d] = n/2 (a + l), where l is the last term.

等差数列：第 n 项 uₙ = a + (n - 1)d ；前 n 项和 Sₙ = n/2 [2a + (n - 1)d] = n/2 (a + l)，其中 l 为末项。

sequence: nth term uₙ = arⁿ⁻¹ ; sum of first n terms Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1. If |r| < 1, the sum to infinity exists: S∞ = a/(1 - r).

等比数列：第 n 项 uₙ = arⁿ⁻¹ ；前 n 项和 Sₙ = a(1 - rⁿ)/(1 - r)，r ≠ 1。若 |r| < 1，无穷和存在：S∞ = a/(1 - r)。

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6. Binomial Expansion | 二项式展开

nteger n, (a + b)ⁿ = Σ (from k=0 to n) ⁿCₖ aⁿ⁻ᵏ bᵏ, where the binomial coefficient ⁿCₖ = n! / (k!(n - k)!). In particular, (1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + [n(n-1)(n-2)/3!] x³ + ...

对于正整数 n，(a + b)ⁿ = Σ（k=0 到 n）ⁿCₖ aⁿ⁻ᵏ bᵏ，其中二项式系数 ⁿCₖ = n! / (k!(n - k)!)。特别地，(1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + [n(n-1)(n-2)/3!] x³ + ...

When n is fractional or negative and |x| < 1, the expansion (1 + x)ⁿ is valid as an infinite series using the same pattern of coefficients: 1 + nx + n(n-1)x²/2! + ...

当 n 为分数或负数且 |x| < 1 时，(1 + x)ⁿ 展开为无穷级数，系数规律相同：1 + nx + n(n-1)x²/2! + ...

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7. Calculus – Differentiation | 微积分 – 微分

The power rule for differentiation: if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. This extends to sums and constant multiples: d/dx [c f(x)] = c f'(x) ; d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

幂函数微分法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。该法则适用于线性组合：d/dx [c f(x)] = c f'(x) ；d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。

Equation of the tangent at (x₁, y₁): y - y₁ = m (x - x₁) where m = f'(x₁). Stationary points occur where f'(x) = 0; use the second derivative f''(x) to classify them: f''(x) > 0 indicates a minimum, f''(x) < 0 indicates a maximum.

点 (x₁, y₁) 处的切线方程：y - y₁ = m (x - x₁)，其中 m = f'(x₁)。驻点出现在 f'(x)=0 处；用二阶导数 f''(x) 判定：f''(x) > 0 为极小值，f''(x) < 0 为极大值。

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8. Calculus – Integration | 积分

tion: ∫ xⁿ dx = (xⁿ⁺¹)/(n + 1) + C, for n ≠ -1. The constant of integration C is required.

不定积分的幂法则：∫ xⁿ dx = (xⁿ⁺¹)/(n + 1) + C，n ≠ -1。必须加上积分常数 C。

alculate the net area between the curve and the x-axis: ∫ₐᵇ f(x) dx = F(b) - F(a), where F'(x) = f(x). Area between a curve and the x-axis is given by ∫ₐᵇ |f(x)| dx if the graph goes below the axis.

定积分计算曲线与 x 轴之间的净面积：∫ₐᵇ f(x) dx = F(b) - F(a)，其中 F'(x) = f(x)。若图像在 x 轴下方，曲线与 x 轴之间的面积通过 ∫ₐᵇ |f(x)| dx 求出。

ocity v = ds/dt, acceleration a = dv/dt; displacement s = ∫ v dt, and velocity v = ∫ a dt.

运动学应用：速度 v = ds/dt，加速度 a = dv/dt；位移 s = ∫ v dt，速度 v = ∫ a dt。

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9. Vectors | 向量

vector in 2D can be written as ai + bj or as a column vector [a ; b]. Magnitude (length): |ai + bj| = √(

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