📚 GCSE Maths: Core Formulas and Problem-Solving Tips | GCSE数学核心公式梳理与解题要点
This revision guide brings together the essential formulas and problem-solving strategies for GCSE Mathematics. It follows the major topic areas of the specification, helping you consolidate your knowledge and apply it with confidence.
本复习指南汇总了 GCSE 数学中最重要的公式与解题策略，覆盖大纲主要板块，帮助你巩固知识并自信运用。
1. Number and Operations | 数与运算
Standard form is used for very large or very small values. A number is in standard form when written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
标准形式用于表示非常大或非常小的数值。一个数写成 A × 10ⁿ 即为标准形式，其中 1 ≤ A < 10，n 是整数。
When multiplying numbers in standard form, multiply the front parts and add the powers of 10:
标准形式相乘时，先将前面的数相乘，再将 10 的指数相加：
(A × 10ᵐ) × (B × 10ⁿ) = (A × B) × 10ᵐ⁺ⁿ
When adding or subtracting, first rewrite both numbers to the same power of 10.
加减标准形式时，应先把两个数化为相同的 10 的幂次。
Laws of indices are fundamental throughout GCSE algebra:
指数运算律贯穿 GCSE 代数的始终：
xᵃ × xᵇ = xᵃ⁺ᵇ, xᵃ ÷ xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ, x⁰ = 1, x⁻ⁿ = 1/xⁿ
Use prime factor trees to express a number as a product of primes. For HCF, take the lowest power of each common prime; for LCM, take the highest power of each prime found in either number.
用质因数树把一个数写成质数乘积。求 HCF 时取共同质数的最低次幂；求 LCM 时取所有质数的最高次幂。
Remember that √x = x^½ and ∛x = x^⅓. Fractional powers correspond to roots.
记住 √x = x^½，∛x = x^⅓。分数指数对应着开方运算。
2. Algebraic Expressions and Factorising | 代数表达式与因式分解
Expanding brackets requires multiplying every term inside one bracket by every term inside the other bracket.
展开括号时，需要将一个括号中的每一项与另一个括号中的每一项相乘。
(a + b)(c + d) = ac + ad + bc + bd
Difference of two squares and perfect squares are key factorising patterns:
平方差与完全平方是重要的因式分解模式：
a² − b² = (a + b)(a − b), (a ± b)² = a² ± 2ab + b²
To factorise x² + p x + q, find two numbers whose sum is p and whose product is q.
要分解 x² + p x + q，找到两个数，使它们的和为 p、积为 q。
Always look for a common factor first. If the coefficient of x² is not 1, try factorising by grouping or use the ac-method.
先寻找公因式。如果 x² 的系数不为 1，可尝试分组分解或使用 ac 法。
Completing the square: x² + b x = (x + b/2)² − (b/2)². This is useful when solving quadratics or finding turning points.
配方法：x² + b x = (x + b/2)² − (b/2)²。在解一元二次方程或求函数最值时十分有效。
3. Equations and Inequalities | 方程与不等式
Linear equations are solved by using inverse operations to isolate the variable. Whatever you do to one side, do to the other.
解一次方程时，用逆运算来隔离未知数。对一边做的操作必须同时对另一边进行。
For a quadratic equation a x² + b x + c = 0, the solutions are given by the quadratic formula:
对于二次方程 a x² + b x + c = 0，其解由求根公式给出：
x = (−b ± √b² − 4ac) / 2a
The discriminant b² − 4ac tells you the number of real roots: if positive, two distinct roots; if zero, one repeated root; if negative, no real roots.
判别式 b² − 4ac 决定实数根的个数：大于 0 时有两个不同实根；等于 0 时有一个重根；小于 0 时没有实根。
If the quadratic factorises, set each bracket equal to zero instead of using the formula.
如果二次式能因式分解，则令每个因式等于零，而不必套用公式。
For simultaneous equations, use substitution when one equation is linear and the other is quadratic; use elimination when both are linear.
解联立方程组时，若一个为一次方程、另一个为二次方程，用代入法；若都为一次方程，用消元法。
Inequalities are solved like equations, but reverse the inequality sign when multiplying or dividing by a negative number.
解不等式与解方程类似，但当乘或除以负数时，必须改变不等号方向。
4. Functions and Graphs | 函数与图像
The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept.
直线方程为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
m = (y₂ − y₁) / (x₂ − x₁)
Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to −1, so m₁ × m₂ = −1.
平行线的斜率相同。互相垂直的直线斜率乘积为 −1，即 m₁ × m₂ = −1。
The quadratic graph y = a x² + b x + c is a parabola. Its turning point can be found by completing the square or using x = −b/(2a).
二次函数 y = a x² + b x + c 的图像是抛物线。可通过配方法或 x = −b/(2a) 求其顶点。
The graph of y = a/x is a hyperbola with two symmetrical branches; it is undefined at x = 0.
y = a/x 的图像是双曲线，有两个对称分支；在 x = 0 处无定义。
To solve an equation graphically, draw both curves and read the x-coordinates of their intersection points.
用图像法解方程时，画出两条曲线，读取交点横坐标即可。
5. Ratio, Proportion and Percentages | 比率、比例与百分比
Direct proportion means y = kx; inverse proportion means y = k/x, where k is the constant of proportionality.
正比例表示 y = kx；反比例表示 y = k/x，其中 k 为比例常数。
y = kx, y = k/x
To share a quantity in a ratio, add the parts of the ratio to find the total number of parts, then divide the quantity by that total.
按比例分配时，先将比的各项相加得到总份数，再用总量除以总份数。
Percentage change = (change / original) × 100%. If the change is an increase, add to 100%; if a decrease, subtract from 100%.
百分比变化 =（变化量 ÷ 原值）× 100%。若为增加，则在 100% 基础上加；若为减少，则从 100% 中减。
Compound growth and decay use repeated multipliers: final = P × (1 ± r/100)ⁿ, where P is the initial amount, r is the percentage rate per period, and n is the number of periods.
复利增长与衰减使用重复乘数：最终值 = P × (1 ± r/100)ⁿ，其中 P 为初始量，r 为每期百分比，n 为期数。
For repeated percentage problems, always apply the multiplier each time rather than adding percentages.
处理连续百分比问题时，应每次使用乘数，而不是直接相加百分数。
6. Geometry: Angles, Polygons and Circles | 几何：角度、多边形与圆
The angle relationships in polygons are essential for many problems.
多边形中的角度关系是解决许多问题的关键。
sum of interior angles = (n − 2) × 180°, exterior angle of regular n-gon = 360° / n
Circle measurements use π:
圆的计算使用 π：
C = 2πr = πd, A = πr²
The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.
圆心角是对同弧上的圆周角的 2 倍。
Angles in the same segment are equal. Opposite angles in a cyclic quadrilateral sum to 180°.
同弦上的圆周角相等。圆内接四边形的对角和为 180°。
A tangent to a circle is perpendicular to the radius at the point of contact. The angle between a tangent and a chord equals the angle in the alternate segment.
切线与过切点的半径垂直。切线与弦的夹角等于弦切角定理中的圆周角。
When using circle theorems, mark known angles on the diagram first; look for isosceles triangles made by two radii.
运用圆定理时，先在图上标出已知角；注意两条半径构成的等腰三角形。
7. Measurement: Area, Volume and Surface Area | 测量：面积、体积与表面积
Memorise the standard area formulas for flat shapes, then extend them to prisms and solids.
熟记平面图形的面积公式，再将其推广到棱柱和立体图形。
triangle = ½bh, parallelogram = bh, trapezium = ½(a + b)h
For volumes, we use the appropriate formula based on the solid.
关于体积，应根据立体类型选择相应公式。
prism = area of cross-section × length, &
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