📚 Year 10 WJEC Further Maths: Formula & Theorem Quick Reference | 进阶数学公式定理速查手册
This bilingual quick reference handbook collects the essential formulae and theorems for the Year 10 WJEC Further Mathematics syllabus. It is designed to help you review key results in algebra, polynomials, coordinate geometry, trigonometry, sequences, calculus, vectors, matrices, probability and mechanics in one place. Each entry is presented in English first, followed by a centred, bold formula and its Chinese explanation, so that bilingual learners can move between the two languages without losing the mathematical meaning. Use it as a last-minute revision sheet before a topic test, or as a checklist while you work through past papers.
这本双语速查手册汇集了 Year 10 WJEC 进阶数学大纲中的必备公式和定理，帮助你在一个页面内快速复习代数、多项式、坐标几何、三角学、数列、微积分、向量、矩阵、概率和力学等关键内容。每个条目先用英文陈述，再以居中加粗的公式和中文解释呈现，让双语学习者在中英文之间切换时不会丢失数学含义。你可以在章节小测前把它当作考前速览，也可以在做真题时当作核对清单使用。
1. Algebraic Basics | 代数基础
Indices obey three core laws that you should be able to quote instantly in an exam. When you multiply powers with the same base you add the exponents, when you divide them you subtract, and when you raise a power to a further power you multiply. These laws extend to zero, negative and fractional indices, which appear constantly in WJEC questions on surds, standard form and algebraic simplification.
指数有三条核心运算法则，考试中必须能脱口而出。同底数幂相乘，指数相加；同底数幂相除，指数相减；幂的乘方，指数相乘。这些法则同样适用于零指数、负指数和分数指数，它们频繁出现在 WJEC 关于根式、标准形式与代数化简的题目中。
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ
同底数幂的乘、除与乘方法则：指数分别相加、相减、相乘。
The three special cases below extend the same laws to every rational exponent. Remember that a negative index never makes the answer negative; it produces a reciprocal.
下面三个特例把同样的法则推广到任意有理指数。请记住：负指数不会让结果变成负数，它只是产生一个倒数。
a⁰ = 1 (a ≠ 0) a⁻ⁿ = 1 ÷ aⁿ a1/n = ⁿ√a am/n = ⁿ√aᵐ
零指数、负指数与分数指数：a⁰ = 1；a⁻ⁿ 等于 aⁿ 的倒数；a1/n 表示 a 的 n 次方根，am/n 表示先乘方再开方。
Expanding a single bracket means multiplying every term inside the bracket by the factor outside, and the distributive law is the justification for the whole process.
展开单项式乘多项式，就是把括号外的因式乘到括号内的每一项，其依据是乘法分配律。
a(b + c) = ab + ac
单项式乘多项式：把括号外的因式乘到括号内的每一项。
Two standard products should be memorised in both directions, because factorising is simply the reverse reading of the same identity.
下面两个常用乘法公式必须正反两用都熟练，因为因式分解本质上就是把恒等式反向读取。
(a + b)(a - b) = a² - b²
平方差公式：两数之和与两数之差的乘积等于两数的平方差。
(a ± b)² = a² ± 2ab + b²
完全平方公式：二项式平方展开后为"首平方、尾平方、乘积两倍放中央"。
2. Expanding, Factorising and Algebraic Fractions | 展开、因式分解与代数分式
ucts in a 2 × 2 grid before collecting like terms, which reduces sign errors dramatically.
两个括号相乘时，第一个括号中的每一项都要与第二个括号中的每一项相乘。一个可靠的考试习惯是先把四个乘积写进 2 × 2 的网格里，再合并同类项，这样能大幅减少符号错误。
(a + b)(c + d) = ac + ad + bc + bd
双括号展开法则：逐项相乘，再合并同类项。
Factorising a quadratic with leading coefficient 1 is the reverse of expanding. You look for two numbers whose product is the constant term and whose sum is the coefficient of x.
首项系数为 1 的二次式因式分解，其实就是展开的逆过程。你需要找到两个数，它们的积等于常数项，它们的和等于一次项系数。
x² + (p + q)x + pq = (x + p)(x + q)
十字相乘法（首项系数为 1）：找两个数 p、q，使 p + q 等于一次项系数，pq 等于常数项。
Completing the square rewrites the same expression in vertex form, which is the fastest way to locate a turning point or to prove that a quadratic is always positive.
配方法把同一个表达式改写为顶点式，这是求二次函数顶点、或证明二次式恒为正的最快方法。
x² + bx + c = (x + b/2)² + c - b²/4
配方法：加上并减去一次项系数一半的平方，得到顶点式 (x + b/2)² + c - b²/4。
Algebraic fractions follow exactly the same rules as numerical fractions. To add or subtract, use a common denominator; to divide, multiply by the reciprocal; and always cancel only after factorising.
代数分式的运算法则与数字分数完全相同。加减需要先通分；除法要乘以倒数；并且一定要先因式分解，再约分。
a/b + c/d = (ad + bc) ÷ bd (a/b) ÷ (c/d) = (a/b) × (d/c)
分式加减与除法：通分后分子相加；除以一个分式等于乘以它的倒数。
3. Quadratic Equations and the Discriminant | 二次方程与判别式
t factorise with integers. Always rearrange to standard form and identify a, b and c carefully before substituting, and remember that the whole numerator is divided by 2a.
只要方程化为标准形式 ax² + bx + c = 0，二次公式就能求出它的根，包括那些无法用整数因式分解的方程。代入之前务必先整理成标准形式，并仔细辨认 a、b、c，同时牢记整个分子都要除以 2a。
x = (-b ± √b² - 4ac) ÷ (2a)
二次公式给出 ax² + bx + c = 0 的两个根。
The expression under the square root is called the discriminant, and its sign tells you how many real roots the equation has before you do any further work.
根号下的表达式称为判别式，它的符号能在你进一步计算之前，直接告诉你方程有多少个实根。
Δ = b² - 4ac
判别式 Δ = b² - 4ac 决定实根的个数。
| Value of Δ | 判别式的值 | Nature of roots | 根的性质 |
| Δ > 0 | Two distinct real roots | 两个不相等的实根 |
| Δ = 0 | One repeated real root | 一个二重实根 |
| Δ < 0 | No real roots | 没有实根 |
When a quadratic does have real roots α and β, you can recover both the coefficient of x and the constant term from their sum and product. This shortcut is very common in WJEC further mathematics proof questions.
当二次方程有实根 α 和 β 时，你可以由它们的和与积反推出一次项系数和常数项。这个技巧在 WJEC 进阶数学的证明题中非常常见。
α + β = -b ÷ a αβ = c ÷ a
韦达定理：两根之和等于 -b/a，两根之积等于 c/a。
4. Polynomials: Remainder and Factor Theorems | 多项式：余数定理与因式定理
A polynomial is an expression built from non-negative integer powers of x. Substituting a number into a polynomial is called evaluating it, and this simple operation is the basis of both the remainder theorem and the factor theorem.
多项式是由 x 的非负整数次幂构成的表达式。把某个数值代入多项式求值，这一简单操作正是余数定理与因式定理的共同基础。
The remainder theorem states that when a polynomial f(x) is divided by (x - a), the remainder equals f(a). You therefore never need long division to find a remainder.
余数定理指出：多项式 f(x) 除以 (x - a) 时，余数等于 f(a)。因此求余数完全不需要做长除法。
f(x) = (x - a)Q(x) + f(a)
余数定理：f(x) 除以 (x - a) 的余数为 f(a)，其中 Q(x) 为商式。
The factor theorem is the special case in which the remainder is zero. It gives a fast test for whether a linear expression is a factor, and it is the standard first step in factorising a cubic.
因式定理是余数为零的特殊情形。它提供了判断某个一次式是否为因式的快速检验方法，也是三次多项式因式分解的标准第一步。
(x - a) is a factor of f(x) ⇔ f(a) = 0
因式定理：(x - a) 是 f(x) 的因式 当且仅当 f(a) = 0。
For longer divisions the identity below is worth memorising, because it explains exactly what each part of a long division calculation represents.
对于较长的除法，下面的恒等式值得记住，因为它清楚说明了长除法中每一部分所代表的含义。
Dividend = Divisor × Quotient + Remainder
被除式 = 除式 × 商式 + 余式。
5. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆
Coordinate geometry questions nearly always begin with two points, so the distance and midpoint formulae should be at your fingertips. Both are direct applications of Pythagoras' theorem and of averaging.
坐标几何题几乎总是从两个点出发，因此距离公式和中点公式必须烂熟于心。它们分别是勾股定理与求平均值的直接应用。
d = √((x₂ - x₁)² + (y₂ - y₁)²)
两点间距离公式：横坐标之差与纵坐标之差的平方和再开方。
M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
中点公式：横坐标取平均，纵坐标取平均。
The gradient measures steepness, and the point-gradient form is the quickest way to write down a line once you know one point and the slope. Parallel lines share the same gradient; perpendicular lines have gradients whose product is -1.
斜率衡量直线的陡峭程度；已知一点和斜率时，点斜式是写出直线方程最快的方法。平行直线斜率相同；垂直直线的斜率之积为 -1。
m = (y₂ - y₁) ÷ (x₂ - x₁) y - y₁ = m(x - x₁)
斜率公式与点斜式方程。
Parallel: m₁ = m₂ Perpendicular: m₁ × m₂ = -1
平行与垂直的判定条件。
A circle centred at the origin is the set of points at a fixed distance from the centre, and shifting the centre simply changes the constants inside the brackets. Expanding the standard form gives the general equation, from which the centre and radius can be read off by completing the square.
圆心在原点的圆，是到圆心距离等于定长的点的集合；圆心平移只会改变括号内的常数。把标准式展开就得到一般式，通过配方即可读出圆心与半径。
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