📚 Year 9 Edexcel Further Maths: Quick-Reference Formula & Theorem Handbook | 九年级Edexcel进阶数学：公式定理速查手册
This quick-reference handbook summarises all the essential formulas, theorems and key facts you need for Year 9 Edexcel Further Mathematics. Each topic is presented with clear statements in English, immediately followed by the Chinese translation, helping you revise efficiently and build a solid foundation for GCSE success.
这本速查手册汇总了九年级 Edexcel 进阶数学所需的全部核心公式、定理与关键知识点。每个主题都先用英文清晰陈述，紧接提供中文翻译，帮助你高效复习，为 GCSE 的成功打下坚实基础。
1. Algebraic Expressions & Factorisation | 代数表达式与因式分解
y). To expand brackets, multiply each term inside by the term outside: a(b + c) = ab + ac.
代数中我们用字母表示数。项（如 3x²y）包含系数（3）和变量（x, y）。去括号时，将外面的项乘以括号内的每一项：a(b + c) = ab + ac。
)(x + 3) because 2+3=5 and 2×3=6. The difference of two squares: a² - b² = (a + b)(a - b).
二次式常常因式分解为两个括号。例如 x² + 5x + 6 = (x + 2)(x + 3)，因为 2+3=5 且 2×3=6。平方差公式：a² - b² = (a + b)(a - b)。
To factorise a quadratic in the form ax² + bx + c where a ≠ 1, find two numbers that multiply to ac and add to b, then split the middle term and factor by grouping.
当 a ≠ 1 的二次式 ax² + bx + c 因式分解时，找出乘积为 ac 且和为 b 的两个数，再拆分中间项并分组分解。
2. Solving Linear & Quadratic Equations | 解一次方程与二次方程
ons. Always do the same to both sides. For example, 2x - 5 = 9 leads to 2x = 14, so x = 7.
解一次方程时，用逆运算分离变量，两边始终进行相同的操作。如 2x - 5 = 9，得 2x = 14，因此 x = 7。
be solved by factorising. Set each bracket equal to zero. If factorising is not possible, use the quadratic formula:
二次方程 ax² + bx + c = 0 可通过因式分解求解，令每个括号等于 0。若无法因式分解，则使用求根公式：
x = [ -b ± √b² - 4ac ] / (2a)
The expression under the square root, b² - 4ac, is called the discriminant. It tells us the number of real roots:
平方根号下的 b² - 4ac 称为判别式，它告诉我们实根的数量：
- If b² - 4ac > 0, two distinct real roots. | 若 > 0，有两个不相等的实根。
- If b² - 4ac = 0, exactly one real root (repeated). | 若 = 0，有一个实根（重根）。
- If b² - 4ac < 0, no real roots. | 若 < 0，没有实根。
3. Inequalities | 不等式
An inequality compares two expressions. Solve like an equation, but remember: multiplying or dividing by a negative number reverses the inequality sign.
不等式用于比较两个表达式。解法同方程，但要注意：乘以或除以负数时，不等号方向改变。
solutions on a number line with open or closed circles, and using set notation like x ≤ 3.
例如 -2x < 8 变为 x > -4。解集可用数轴表示，空心圆表不包含，实心圆表包含，或用集合记号如 x ≤ 3。
Quadratic inequalities such as x² - 4x + 3 > 0 are solved by sketching the graph or using sign diagrams. The region above the x-axis (y > 0) gives the solution.
二次不等式如 x² - 4x + 3 > 0 可通过草图或符号图求解，图像在 x 轴上方（y > 0）的区域即为解集。
4. Straight Line Graphs | 直线图像
where m is the gradient and c is the y-intercept. Gradient = (change in y)/(change in x).
直线方程通常写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距。斜率 = (y 的改变量)/(x 的改变量)。
nt. Two lines are parallel if they have the same gradient. Two lines are perpendicular if the product of their gradients equals -1: m₁ × m₂ = -1.
若两直线斜率相同则平行；若斜率乘积为 -1 （m₁ × m₂ = -1）则垂直。
ne given two points (x₁, y₁) and (x₂, y₂), first calculate m, then use y - y₁ = m(x - x₁).
已知两点 (x₁, y₁) 和 (x₂, y₂) 求直线方程，先计算斜率 m，再使用点斜式 y - y₁ = m(x - x₁)。
5. Simultaneous Equations | 联立方程组
Simultaneous equations can be solved by elimination (adding or subtracting equations) or substitution. For two linear equations, align terms and eliminate one variable.
联立方程组可用消元法（加减方程）或代入法求解。对于两个一次方程，对齐同类项后消去一个变量。
For one linear and one quadratic, substitution is usually needed. Replace y from the linear equation into the quadratic, then solve for x and substitute back.
若为一个一次方程加一个二次方程，通常需要代入法。将一次方程的 y 代入二次式中，解出 x 后再回代求 y。
Always check your solutions satisfy both original equations. Graphically, the solution is the intersection point(s) of the lines or curves.
务必检验所得解是否同时满足两个原方程。图像上，解就是直线或曲线的交点。
6. Indices & Surds | 指数与根式
aws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ, am/n = (ⁿ√a)ᵐ.
指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1（a ≠ 0）, a⁻ⁿ = 1/aⁿ, am/n = (ⁿ√a)ᵐ。
A surd is an irrational root, such as √2. Simplify surds: √ab = √a × √b, and rationalise denominators like 1/√2 = √2/2.
根式是无理方根，如 √2。化简根式：√ab = √a × √b，并要将分母有理化，如 1/√2 = √2/2。
To rationalise a denominator with a + b√c, multiply numerator and denominator by its conjugate a - b√c.
若分母为 a + b√c，分子分母同乘其共轭式 a - b√c 来实现有理化。
7. Pythagoras & Trigonometry | 勾股定理与三角学
Pythagoras' triangle, a² + b² = c², where c is the hypotenuse (longest side opposite the right angle).
勾股定理：在直角三角形中，a² + b² = c²，c 为斜边（直角所对的最长边）。
Trigonometric ratios for a right-angled triangle with angle θ:
直角三角形中角 θ 的三角比：
| sin θ = opposite/hypotenuse | 正弦 = 对边/斜边 |
| cos θ = adjacent/hypotenuse | 余弦 = 邻边/斜边 |
| tan θ = opposite/adjacent = sinθ/cosθ | 正切 = 对边/邻边 = sinθ/cosθ |
= 1/2, tan 45° = 1, etc. Pythagoras and trigonometry are used to solve 2D and 3D problems.
常考的准确三角值：sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1 等。勾股定理和三角比用于解决二维和三维问题。
8. Area, Volume & Surface Area | 面积、体积与表面积
area = ½ × base × height; Parallelogram area = base × height; Trapezium area = ½(a + b)h.
二维图形核心公式：矩形面积 = 长×宽；三角形面积 = ½×底×高；平行四边形面积 = 底×高；梯形面积 = ½(上底+下底)×高。
ce = 2πr or πd. Sector arc length = (θ/360)×2πr, sector area = (θ/360)×πr² (θ in degrees).
圆面积 = πr²，周长 = 2πr 或 πd。扇形弧长 = (θ/360)×2πr，扇形面积 = (θ/360)×πr² (θ 为度数)。
curved surface area = 2πrh. Pyramid volume = ⅓ × base area × height; cone volume = ⅓πr²h.
棱柱体积 = 截面积 × 长。圆柱体积 = πr²h，侧面积 = 2πrh。棱锥体积 = ⅓ × 底面积 × 高；圆锥体积 = ⅓πr²h。
a sphere, surface area = 4πr², volume = 4/3 πr³. These may appear in advanced Year 9 work.
球体表面积 = 4πr²，体积 = 4/3 πr³。这些可能出现在九年级进阶内容中。
9. Circles – Theorems & Angles | 圆 —— 定理与角度
Angle at the centre is twice the angle at the circumference subtended by the same arc. Angles in the same segment are equal.
圆心角等于同一弧所对圆周角的两倍。同弧上的圆周角相等。
The angle in a semicircle is a right angle (90°). A tangent to a circle is perpendicular to the radius at the point of contact.
半圆所对的圆周角为直角（90°）。圆的切线垂直于过切点的半径。
Opposite angles of a cyclic quadrilateral add up to 180°. The perpendicular from the centre to a chord bisects the chord.
圆内接四边形对角互补（和为 180°）。从圆心垂直于弦的线平分该弦。
Tangent-tangent lengths from an external point are equal. Alternate segment theorem: angle between tangent and chord equals angle in the alternate segment.
从圆外一点引出的两条切线长相等。弦切角定理：切线与弦的夹角等于弦所对的圆周角。
10. Sequences | 数列
t difference d. nth term formula: aₙ = a₁ + (n-1)d, or in linear form: aₙ = dn + (a₁ - d).
等差（线性）数列具有常数差 d。第 n 项公式：aₙ = a₁ + (n-1)d，或写成一次式：aₙ = dn + (a₁ - d)。
A quadratic sequence has a second difference that is constant. The nth term takes the form an² + bn + c. Find a by halving the second difference, then subtract to find b and c.
二次数列的二阶差为常数。其第 n 项形如 an² + bn + c。通过将二阶差除以 2 得到 a，再分别代值求 b 和 c。
h term: aₙ = a₁ × rⁿ⁻¹. Year 9 may cover simple geometric growth or exponential sequences.
等比（几何）数列以常数公比 r 连乘。第 n 项：aₙ = a₁ × rⁿ⁻¹。九年级可能接触简单的几何增长或指数数列。
11. Data & Statistics | 数据与统计
e value when data are ordered; mode is the most frequent value. Range = maximum - minimum.
平均数 = 总和 / 个数。中位数为排序后中间的值；众数是出现频率最高的值。全距 = 最大值 - 最小值。
le range (IQR = upper quartile - lower quartile). Box plots display this summary visually.
累积频数图用于求中位数与四分位距 (IQR = 上四分位数 - 下四分位数)。箱形图直观地展示这些统计量。
Scatter graphs show correlation: positive, negative or none. A line of best fit can be used to estimate values. Correlation does not imply causation.
散点图显示相关性：正相关、负相关或无相关。最佳拟合线可用于估算。相关性不代表因果关系。
12. Probability | 概率
of favourable outcomes) / (total number of equally likely outcomes). It ranges from 0 (impossible) to 1 (certain).
事件的概率 P = 有利结果数 / 等可能结果总数。概率范围从 0（不可能）到 1（必然）。
clusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).
互斥事件满足 P(A 或 B) = P(A) + P(B)。独立事件满足 P(A 且 B) = P(A) × P(B)。
tiplying along branches and adding at the ends. Conditional probability is introduced: P(A|B) = P(A and B)/P(B).
树状图用于计算组合事件的概率，沿分支相乘，结束时相加。条件概率引入：P(A|B) = P(A 且 B)/P(B)。
umber of trials. Always check that probability answers are expressed as fractions, decimals or percentages as required.
期望频次 = 概率 × 试验次数。注意按题目要求将概率答案以分数、小数或百分比给出。
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