📚 IGCSE Edexcel Mathematics: Mastering Algebra and Geometry | IGCSE Edexcel 数学：代数与几何精讲
Algebra and geometry form the backbone of the Edexcel IGCSE Mathematics syllabus. This revision guide focuses on the key techniques you need to solve equations, manipulate expressions, and apply geometric reasoning under exam conditions.
代数与几何是 Edexcel IGCSE 数学课程的两大支柱。本复习指南将聚焦核心解题技巧，帮助你在考试条件下熟练求解方程、变形表达式，并运用几何推理答题。
1. Expanding and Factorising Expressions | 展开与因式分解
Expanding brackets requires multiplying every term inside the bracket by the term outside. For double brackets, use the FOIL method: First, Outer, Inner, Last.
展开括号时，需要用括号外的每一项乘以括号内的每一项。对于双重括号，使用 FOIL 方法：首项（First）、外项（Outer）、内项（Inner）、末项（Last）。
(x + a)(x + b) = x² + (a + b)x + ab
Factorising is the reverse process. Always look for a common factor first, then consider difference of two squares or quadratic factorisation.
因式分解是展开的逆过程。先找公因式，再考虑平方差公式或二次三项式的分解。
Difference of two squares: x² − 9 = (x + 3)(x − 3)
平方差公式：x² − 9 = (x + 3)(x − 3)
Common factor: 6x² + 3x = 3x(2x + 1)
提取公因式：6x² + 3x = 3x(2x + 1)
Quadratic factorising: x² + 5x + 6 = (x + 2)(x + 3)
二次三项式因式分解：x² + 5x + 6 = (x + 2)(x + 3)
2. Solving Linear and Quadratic Equations | 解线性方程与二次方程
Linear equations are solved by performing inverse operations on both sides. For quadratic equations, factorising is usually the fastest method when the expression factorises neatly.
解线性方程时，两边同时进行逆运算。对于二次方程，当表达式能简洁分解时，因式分解通常是最快的方法。
x² + bx + c = 0, factorise into (px + q)(rx + s) = 0, then set each bracket equal to zero.
If the quadratic cannot be factorised, use the quadratic formula:
若二次方程无法因式分解，则使用求根公式：
x = (−b ± √b² − 4ac) / 2a
Example: x² − 3x − 4 = 0 → (x − 4)(x + 1) = 0 → x = 4 or x = −1
示例：x² − 3x − 4 = 0 → (x − 4)(x + 1) = 0 → x = 4 或 x = −1
Check your solutions by substituting back into the original equation.
将解代回原方程进行检验。
3. Simultaneous Equations | 联立方程组
Edexcel IGCSE often tests simultaneous equations with one linear and one quadratic equation. Solve by substitution: rearrange the linear equation to make x or y the subject, then substitute into the quadratic.
Edexcel IGCSE 常考一个线性方程与一个二次方程组成的联立方程组。采用代入法：将线性方程改写为用 x 或 y 表示，再代入二次方程。
y = 2x + 1 and y = x² − x + 3 → x² − x + 3 = 2x + 1
Then solve the resulting quadratic. You may obtain two solutions for x, and correspondingly two solutions for y.
然后解所得二次方程。x 可能有两个解，相应地 y 也有两个解。
| Method | 方法 | Use when | 适用情况 |
| Elimination 消元法 | Both equations linear 两个方程均为线性 |
| Substitution 代入法 | One equation is quadratic 其中一个为二次方程 |
4. Algebraic Fractions and Indices | 代数分式与指数
Algebraic fractions follow the same rules as numerical fractions. Find a common denominator before adding or subtracting, and simplify by cancelling common factors.
代数分式与数字分式的运算规则相同。加减前先找公分母，并通过约分公因式进行化简。
1/(x+1) + 2/(x−1) = (x−1 + 2(x+1)) / ((x+1)(x−1)) = (3x+1) / (x²−1)
Indices rules are essential. Remember x⁰ = 1, x⁻ⁿ = 1/xⁿ, and xm/n = ⁿ√xm.
指数法则至关重要。牢记 x⁰ = 1、x⁻ⁿ = 1/xⁿ，以及 xm/n = ⁿ√xm。
xa × xb = xa+b
同底数幂相乘，指数相加。
xa ÷ xb = xa−b
同底数幂相除，指数相减。
(xa)b = xab
幂的乘方，指数相乘。
5. Straight Line Graphs and Gradients | 直线图像与斜率
x + c, where m is the gradient and c is the y-intercept. The gradient describes how steep the line is.
直线方程的标准形式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。斜率表示直线的倾斜程度。
m = (y₂ − y₁) / (x₂ − x₁)
If two lines are parallel, their gradients are equal. If two lines are perpendicular, the product of their gradients is −1.
若两条直线平行，则斜率相等。若两条直线垂直，则斜率乘积为 −1。
Parallel: m₁ = m₂
平行：m₁ = m₂
Perpendicular: m₁ × m₂ = −1
垂直：m₁ × m₂ = −1
6. Circle Theorems and Angle Rules | 圆定理与角度法则
Circle theorems are a favourite in Edexcel IGCSE papers. You must know the following: the angle at the centre is twice the angle at the circumference; the angle in a semicircle is 90°; opposite angles in a cyclic quadrilateral sum to 180°.
圆定理是 Edexcel IGCSE 试卷中的常客。你必须掌握以下定理：圆心角是圆周角的两倍；半圆内的圆周角为90°；圆内接四边形对角之和为180°。
Angle at centre = 2 × angle at circumference
Also remember that the tangent to a circle is perpendicular to the radius at the point of contact, and that tangents from an external point are equal in length.
还要记住：圆的切线与切点处的半径垂直，且从圆外一点引出的两条切线长度相等。
| Theorem | 定理 | Key fact | 关键结论 |
| Angle in semicircle 半圆圆周角 | Always 90° 恒为90° |
| Cyclic quadrilateral 圆内接四边形 | Opposite angles sum to 180° 对角和为180° |
| Alternate segment theorem 弦切角定理 | Angle between tangent and chord equals angle in opposite segment 切线与弦的夹角等于相对弧段上的圆周角 |
7. Area and Volume of 2D and 3D Shapes | 二维与三维图形的面积和体积
d volume. Practice converting between units, especially cm³ and m³ (1 m³ = 1,000,000 cm³).
你需要熟记面积和体积的标准公式。多练习单位换算，尤其是 cm³ 与 m³（1 m³ = 1,000,000 cm³）。
Area of trapezium = ½(a + b)h
梯形面积 = ½(a + b)h
Circumference of circle = 2πr, area = πr²
圆周长 = 2πr，面积 = πr²
Volume of cylinder = πr²h
圆柱体积 = πr²h
Volume of sphere = (4/3)πr³
球体积 = (4/3)πr³
For compound shapes, split them into simpler parts and add or subtract their areas or volumes.
对于复杂图形，将其分割为简单的组成部分，然后进行面积或体积的加减。
8. Trigonometry in Right-Angled Triangles | 直角三角形中的三角函数
The trigonometric ratios SOHCAHTOA are essential for solving right-angled triangle problems.
SOHCAHTOA 三个三角函数比值是解决直角三角形问题的关键。
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
For non-right-angled triangles, use the sine rule and cosine rule:
对于非直角三角形，使用正弦定理和余弦定理：
a/sin A = b/sin B = c/sin C
c² = a² + b² − 2ab cos C
Remember to set your calculator to degrees when giving answers in degrees.
当答案使用度时，记得将计算器设定为角度制。
9. Vectors | 向量
Vectors have both magnitude and direction. In IGCSE, vectors are usually written as column vectors or using bold letters like a, b.
向量同时具有大小和方向。在 IGCSE 中，向量通常写作列向量形式，或用粗体字母如 a、b 表示。
a = (3, −2) means 3 units right and 2 units down
To add vectors, add their components. To subtract, subtract components. Multiplying a vector by a scalar multiplies each component.
向量加法按分量相加，减法按分量相减。标量乘以向量时，每个分量都乘以该标量。
If a = (2, 1) and b = (3, 4), then a + b = (5, 5)
若 a = (2, 1)，b = (3, 4)，则 a + b = (5, 5)
3a = (6, 3)
3a = (6, 3)
10. Transformations | 几何变换
Four types of transformations appear in the exam: translation, reflection, rotation, and enlargement. You must describe each transformation fully.
考试中出现四种变换类型：平移、反射、旋转和放大。你必须完整描述每种变换。
| Transformation | 变换 | Required information | 必要信息 |
| Translation 平移 | Vector 向量 |
| Reflection 反射 | Mirror line 对称轴 |
| Rotation 旋转 | Centre, angle, direction 旋转中心、角度、方向 |
| Enlargement 放大 | Centre, scale factor 位似中心、比例因子 |
For enlargement, a negative scale factor produces an image on the opposite side of the centre, rotated 180°.
对于放大，负的比例因子会在中心另一侧生成图像，并旋转180°。
11. Probability and Statistics Basics | 概率与统计基础
Probability is a number between 0 and 1. The sum of probabilities of all mutually exclusive outcomes is 1. For independent events, multiply probabilities.
概率是介于0和1之间的数。所有互斥结果的概率之和为1。对于独立事件，概率相乘。
P(A and B) = P(A) × P(B) if independent
For the mean, median, and mode, know the correct definitions. The interquartile range is the difference between the upper and lower quartiles, and it measures the spread of the middle 50% of the data.
对于平均数、中位数和众数，需要掌握其正确含义。四分位距为上四分位数与下四分位数之差，用于衡量中间50%数据的离散程度。
Mean = sum of values ÷ number of values
平均数 = 数值总和 ÷ 数据个数
Median is the middle value when data is ordered.
中位数是数据按序排列后的中间值。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always show your working clearly. Edexcel awards method marks even if your final answer is wrong. Read the question to see whether it asks for exact values or decimal approximations.
务必清晰展示解题过程。即使最终答案错误，Edexcel 也会给方法分。看清题目要求精确值还是近似小数。
Common mistakes include forgetting to change the sign when moving terms across the equal sign, mixing up the order of operations, and misapplying the cosine rule. Practice past papers and time yourself.
常见错误包括：移项时忘记变号、混淆运算顺序、以及错用余弦定理。多做历年真题并限时训练。
Finally, check your units and round only at the final step to avoid rounding errors.
最后，注意单位换算，只在最后一步四舍五入以避免累积误差。
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