GCSE Maths: Core Formulas and Problem-Solving Tips | GCSE数学核心公式梳理与解题要点

📚 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。

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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。

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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。

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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。

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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。

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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。

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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。

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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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