📚 GCSE Maths: Essential Formulae Handbook | GCSE 数学：必背公式手册
This handbook brings together every essential formula you need for GCSE Mathematics, from number work, indices and algebra to ratio, geometry, trigonometry, vectors, probability and statistics. Each entry is is clearly explained in clear English and then in Chinese, so you can revise bilingually and check your understanding from two directions at once. Use it while working through past papers, when marking your own answers, or as a fast memory check in the final week before the exam.
本手册汇总了 GCSE 数学所需的全部核心公式，覆盖数的运算、指数与代数、比与比例、几何、三角、向量、概率与统计。每一条都先用清晰英文说明，再配中文解释，方便你进行双语复习，并从两个方向检验自己的理解。无论是刷历年真题、批改自己的答案，还是考前最后一周快速回顾，这本手册都能随时派上用场。
1. Fractions, Decimals and Percentages | 分数、小数与百分数
To add or subtract fractions, rewrite both fractions over the common denominator bd, then combine the numerators. The rule is a/b + c/d = (ad + bc)/bd, and subtraction works in exactly the same way with a minus sign in the numerator.
分数加减法要先通分到公分母 bd，再合并分子。公式为 a/b + c/d = (ad + bc)/bd；减法同理，只需把分子中的加号改成减号。
To multiply fractions, multiply the numerators together and the denominators together: a/b × c/d = ac/bd. To divide, multiply by the reciprocal of the second fraction: a/b ÷ c/d = ad/bc.
分数乘法是分子乘分子、分母乘分母：a/b × c/d = ac/bd。分数除法则是乘以除数的倒数：a/b ÷ c/d = ad/bc。
A percentage is simply a fraction with denominator 100, so 35% nd a percentage of an amount, convert the percentage to a decimal multiplier and multiply.
百分数就是分母为 100 的分数，因此 35% = 35/100 = 0.35。求某个量的百分之几时，先把百分数化为小数乘数，再相乘即可。
For a percentage increase, multiply the original amount by the multiplier (1 + r/100), where r is the percentage. For a percentage decrease, use (1 − r/100) instead.
百分数增加时，用原数乘以乘数 (1 + r/100)，其中 r 为百分数；百分数减少时，改用 (1 − r/100)。
Reverse percentages work backwards: divide the new amount by the multiplier that produced it. After an r% increase, original = new ÷ (1 + r/100); after an r% decrease, original = new ÷ (1 − r/100).
逆向百分数是反推过程：用变化后的数值除以当初的乘数。增加 r% 后，原值 = 新值 ÷ (1 + r/100)；减少 r% 后，原值 = 新值 ÷ (1 − r/100)。
Simple interest is calculated only on the original principal: I = P × r/100 × t, where P is the principal, r% the annual rate and t the number of years.
单利只对本金计息：I = P × r/100 × t，其中 P 为本金，r% 为年利率，t 为年数。
Compound interest, compounded annually for n years, uses a repeated multiplier: A = P × (1 + r/100)ⁿ, where A is the final amount, P the principal, r% the annual rate and n the number of years.
复利（按年计息，共 n 年）使用重复乘数：A = P × (1 + r/100)ⁿ，其中 A 为最终金额，P 为本金，r% 为年利率，n 为年数。
A = P × (1 + r/100)ⁿ
2. Indices, and Surds and Standard Form | 指数、根式与标准形式
When multiplying powers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. This only works when the bases are identical.
同底数幂相乘时，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。注意只有底数相同时才能这样运算。
When dividing powers with the same base, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ, provided a ≠ 0.
同底数幂相除时，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ，其中 a ≠ 0。
A power raised to another power multiplies the indices: (aᵐ)ⁿ = aᵐⁿ ful to apply the outer power to every factor inside the bracket, for example (2x³)² = 4x⁶.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。要注意把外层指数作用于括号内每一个因式，例如 (2x³)² = 4x⁶。
Any non-zero number raised to the power zero equals 1: a⁰ = 1 for a ≠ 0. A negative index means a reciprocal: a⁻ⁿ = 1/aⁿ.
任何非零数的零次幂都等于 1：当 a ≠ 0 时，a⁰ = 1。负指数表示取倒数：a⁻ⁿ = 1/aⁿ。
Fractional indices connect to roots: a1/2 = √a and a1/n = ⁿ√a. In general am/n = (ⁿ√a)ᵐ.
分数指数与根式直接对应：a1/2 = √a，a1/n = ⁿ√a。一般地，am/n = (ⁿ√a)ᵐ。
Multiplying surds: √a × √b = √ab. Dividing surds: √a ÷ √b = √a/b for a, b ≥ 0 and b ≠ 0.
根式乘法：√a × √b = √ab。根式除法：√a ÷ √b = √a/b，其中 a、b ≥ 0 且 b ≠ 0。
To rationalise a denominator of the form √b, multiply numerator and denominator by √b: 1/√b = √b/b. For a denominator such as (a + √b), multiply by the conjugate (a − √b).
若分母形如 √b，可分子分母同乘 √b 进行分母有理化：1/√b = √b/b。若分母形如 (a + √b)，则乘以共轭式 (a − √b)。
Standard form writes a number as A × 10ⁿ Multiplying numbers in standard form means multiplying the A values and adding the indices.
标准形式把数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。标准形式相乘时，A 值相乘、指数相加。
3. Algebraic Manipulation and Quadratics | 代数变形与二次式
Expanding two brackets uses every term in the first bracket multiplied by every term in the second: (x + a)(x + b) = x² + (a + b)x + ab.
展开两个括号时，用第一个括号的每一项乘以第二个括号的每一项：(x + a)(x + b) = x² + (a + b)x + ab。
The three special expansions you must know are (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and the difference of two squares a² − b² = (a + b)(a − b).
必须掌握的三个特殊展开式是：(a + b)² = a² + 2ab + b²、(a − b)² = a² − 2ab + b²，以及平方差公式 a² − b² = (a + b)(a − b)。
To factorise a quadratic x² + bx + c, find two numbers that multiply to c and add to b. To factorise ax² + bx + c with a ≠ 1, split the middle term using two numbers that multiply to ac and add to b.
分解 x² + bx + c 时，找两个数，使它们的积为 c、和为 b。分解 a ≠ 1 的 ax² + bx + c 时，找两个积为 ac、和为 b 的数来拆分中间项。
The quadratic formula solves any quadratic ax² + bx + c = 0 when a ≠ 0 and gives x = (−b ± √b² − 4ac) / 2a. Learn it exactly as written, including the ± sign.
二次公式可解任何形如 ax² + bx + c = 0（a ≠ 0）的方程：x = (−b ± √b² − 4ac) / 2a。必须连同 ± 号一字不差地记牢。
x = (−b ± √b² − 4ac) / 2a
The discriminant Δ = b² − 4ac two distinct roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.
判别式 Δ = b² − 4ac 决定实数根的个数：当 Δ > 0 时有两个不同实根，当 Δ = 0 时有一个重根，当 Δ < 0 时没有实根。
Completing the square rewrites a quadratic as (x + p)² + q, where the vertex of the graph lies at (−p, q) and the line of symmetry is x = −p. The turning point of y = ax² + bx + c is at x = −b/2a.
配方法把二次式写成 (x + p)² + q 的形式，图像顶点坐标为 (−p, q)，对称轴为 x = −p。抛物线 y = ax² + bx + c 的转折点横坐标为 x = −b/2a。
Algebraic fractions follow the same rules as numerical fractions, but you must factorise first and cancel common factors rather than individual terms.
代数分式与数字分数遵循相同的运算法则，但要先因式分解，再约去公因式，而不能单独约掉某一项。
4. Equations, Inequalities and Simultaneous Equations | 方程、不等式与联立方程
To solve a linear equation, do the same operation to both sides until the unknown stands alone. Always check your answer by substituting it back into the original equation.
解一元一次方程时，对方程两边做同样的运算，直到未知数单独留在一边。解完一定要把答案代回原方程检验。
A formula can be rearranged by treating it as an equation; whatever you do to one side, you must do to the other. This is called changing the subject of a formula.
公式变形可以当作解方程来处理：对一边做什么运算，就要对另一边做同样的运算。这个过程称为变换公式的主项。
Simultaneous equations can be solved by elimination or substitution. Elimination means matching coefficients and adding or subtracting the equations; substitution means expressing one unknown in terms of the other.
联立方程可用消元法或代入法求解。消元法是让某个未知数的系数相同，再把两式相加减；代入法是把一个未知数用另一个表示后代入。
When solving an inequality, add or subtract as normal, but when you multiply or divide by a negative number you must reverse the inequality sign: if a < b then −a > −b.
解不等式时，加减运算照常进行；但当两边同乘或同除以负数时，不等号方向必须改变：若 a < b，则 −a > −b。
for ≤ or ≥. Two inequalities combined, such as −2 < x ≤ 3, are written as an interval.
在数轴上表示不等式时，< 或 > 用空心圆，≤ 或 ≥ 用实心圆。两个不等式合并，如 −2 < x ≤ 3，可以写成一个区间。
An equation with fractions can be cleared by multiplying every term by the lowest common multiple of the denominators, which removes all fractions in one step.
含分数的方程可以两边同乘各分母的最小公倍数，一步消去所有分母，使计算更简洁。
ing value, then substitute repeatedly until the answer stops changing to the required accuracy.
迭代法用于解无法精确变形的方程。先化为 x = f(x) 的形式，取一个初始值，反复代入，直到结果达到题目要求的精度不再变化。
5. Sequences | 数列
An arithmetic sequence adds the same amount each time. Its nth term is aₙ = a + (n − 1)d, where a is the first term and d the common difference.
等差数列每次加上同一个数。其第 n 项为 aₙ = a + (n − 1)d，其中 a 为首项，d 为公差。
The sum of the first n terms of an arithmetic sequence is Sₙ = n/2 × (2a + (n − 1)d), or equivalently Sₙ = n/2 × (a + l) where l is the last term.
等差数列前 n 项和为 Sₙ = n/2 × (2a + (n − 1)d)，也可写作 Sₙ = n/2 × (a + l)，其中 l 为末项。
A geometric sequence multiplies by the same amount each time. Its nth term is aₙ = a × rⁿ⁻¹, where r is the common ratio.
等比数列每次乘以同一个数。其第 n 项为 aₙ = a × rⁿ⁻¹，其中 r 为公比。
The sum of the first n terms of a geometric sequence is Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1. If r is between −1 and 1, the series converges to S∞ = a/(1 − r).
等比数列前 n 项和为 Sₙ = a(rⁿ − 1)/(r − 1)（r ≠ 1）。当 −1 < r < 1 时，级数收敛于 S∞ = a/(1 − r)。
Special sequences you should recognise on sight include square numbers 1, 4, 9, 16, 25…, cube numbers 1, 8, 27, 64…, triangular numbers 1, 3, 6, 10…, and the Fibonacci sequence 1, 1, 2, 3, 5, 8…
需要一眼认出的特殊数列包括：平方数 1, 4, 9, 16, 25…；立方数 1, 8, 27, 64…；三角数 1, 3, 6, 10…；以及斐波那契数列 1, 1, 2, 3, 5, 8…
For a quadratic sequence, the second differences are constant. The nth term has the form an² + bn + c, where a is half the second difference.
二次数列的二阶差分为常数。其通项形如 an² + bn + c，其中 a 等于二阶差分的一半。
To find a term in a sequence given a term-to-term rule, keep applying the rule. To find the nth term, look for the pattern in the differences and check it against the first few terms.
若给出相邻项规律，只需不断应用该规律即可求后续项。若要求通项公式，则观察差分规律，并用前几项检验结果是否正确。
6. Coordinate Geometry and Straight-Line Graphs | 坐标几何与直线图像
The gradient of a line through two points measures its steepness: m = (y₂ − y₁)/(x₂ − x₁). A positive gradient rises left to right, a negative gradient falls.
过两点的直线斜率表示倾斜程度：m = (y₂ − y₁)/(x₂ − x₁)。斜率为正表示从左向右上升，斜率为负表示下降。
The equation of a straight line is usually written y = mx + c, where m is the gradient and c is the y-intercept. The point-slope form is y − y₁ = m(x − x₁).
直线方程通常写成 y = mx + c，其中 m 为斜率，c 为 y 轴截距。点斜式为 y − y₁ = m(x − x₁)。
Parallel lines have equal gradients: m₁ = m₂. Perpendicular lines have gradients whose product is −1: m₁ × m₂ = −1, so the perpendicular gradient is the negative reciprocal.
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，因此垂直斜率互为负倒数。
The midpoint of two points is found by averaging the coordinates: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点的中点坐标由坐标平均值给出：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
The distance between two points comes from Pythagoras: d = √((x₂ − x₁)² + (y₂ − y₁)²).
两点间距离来自勾股定理：d = √((x₂ − x₁)² + (y₂ − y₁)²)。
he roots are where it crosses the x-axis; the y-intercept is at (0, c); the line of symmetry passes through the vertex.
y = ax² + bx + c 的图像是抛物线。它与 x 轴的交点是方程的根；y 轴截距为 (0, c)；对称轴经过顶点。
Exponential graphs of the form y = kˣ nd decay when 0 < k < 1. All of them pass through (0, 1) and never touch the x-axis.
形如 y = kˣ 的指数函数图像，当 k > 1 时快速上升，当 0 < k < 1 时逐渐衰减。它们都经过点 (0, 1)，且永远不与 x 轴相交。
7. Ratio, Proportion and Rates of Change | 比、比例与变化率
To share an amount in a given ratio, add the parts to find the total number of shares, divide the amount by that total to find one share, then multiply by each part of the ratio.
按比分配时，先把比值各项相加得到总份数，用总量除以总份数得到一份的量，再分别乘以比值中的每一项。
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