IGCSE Mathematics: Core Exam Points in Equations and Algebra | IGCSE数学：方程与代数核心考点梳理

📚 IGCSE Mathematics: Core Exam Points in Equations and Algebra | IGCSE数学：方程与代数核心考点梳理

In IGCSE Mathematics, a solid grasp of equations and algebra is essential for success. This article summarises the core exam points, presenting clear explanations and worked examples that reflect the style of IGCSE questions.

在IGCSE数学中，扎实掌握方程与代数是取得好成绩的关键。本文系统梳理核心考点，通过清晰的讲解和典型例题，帮助考生高效复习、从容应考。

1. Algebraic Expressions and Notation | 代数表达式与符号

An algebraic expression combines numbers, variables (or letters) and operations. For example, in the expression 3x² + 2x – 5, the number 3 is the coefficient of x², 2 is the coefficient of x, and -5 is called the constant term.

代数表达式由数字、变量（字母）和运算符号组合而成。例如，在表达式3x² + 2x – 5中，3是x²的系数，2是x的系数，-5称为常数项。

Terms that have exactly the same variables raised to the same powers are called like terms. Only like terms can be added or subtracted.

所含字母及对应指数完全相同的项称为同类项。只有同类项才能直接相加减。

4x² + 3x² = 7x², but 4x² + 3x cannot be simplified further.

4x² + 3x² = 7x²，但4x² + 3x不能再合并。

2. Expanding and Factorising | 展开与因式分解

Expanding brackets means removing them by multiplication. The distributive law states: a(b + c) = ab + ac. For double brackets, every term in the first bracket multiplies every term in the second bracket.

展开括号是指通过乘法去掉括号。分配律为：a(b + c) = ab + ac。对于两个括号相乘，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。

(x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6

Factorising is the reverse of expanding: writing an expression as a product of factors. Always look for a common factor first. For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b.

因式分解是展开的逆运算：将表达式写成几个因式相乘的形式。首先要寻找公因式。对于x² + bx + c形式的二次三项式，要找两个数，它们的乘积为c，和为b。

x² + 5x + 6 = (x + 2)(x + 3)

Another special case is the difference of two squares: a² – b² = (a + b)(a – b).

另一个特殊情形是平方差公式：a² – b² = (a + b)(a – b)。

3. Linear Equations | 线性方程

A linear equation contains only terms with the variable raised to the power 1. To solve it, use inverse operations to isolate the variable. What you do to one side, you must do to the other.

线性方程中变量指数最高为1。解线性方程时，利用逆运算将变量单独隔离，且等式两边必须进行相同的操作。

Solve 3x + 5 = 14.

解方程3x + 5 = 14。

Subtract 5 from both sides: 3x = 9. Then divide both sides by 3: x = 3. Always check by substituting back into the original equation.

两边同时减5：3x = 9。再两边同时除以3：x = 3。记得将解代回原方程进行检验。

When the unknown appears on both sides, collect the variable terms on one side and the constants on the other. If brackets are present, expand them first.

当未知数同时出现在两边时，先将含未知数的项移到一边，常数项移到另一边。若有括号，先展开括号。

4. Simultaneous Equations | 联立方程组

Simultaneous equations are two or more equations with the same unknown variables. In IGCSE, you often solve a pair of linear equations in two variables using elimination or substitution.

联立方程组是指含有相同未知数的两个或多个方程。在IGCSE中，通常用消元法或代入法解二元一次方程组。

2x + y = 7
x – y = 2

Using elimination, add the two equations to eliminate y: 3x = 9, so x = 3. Substitute x = 3 into x – y = 2, giving 3 – y = 2, so y = 1. The solution is (3, 1).

使用消元法，将两个方程相加消去y：3x = 9，因此x = 3。将x = 3代入x – y = 2，得3 – y = 2，故y = 1。解为(3, 1)。

Substitution is useful when one equation has a single variable as subject. Rearrange that equation, then substitute into the other.

代入法适用于其中一个方程已经表示出某个变量。先将该变量表达式代入另一个方程即可。

5. Quadratic Equations | 二次方程

A quadratic equation has the general form ax² + bx + c = 0. You can solve it by factorising, completing the square, or using the quadratic formula.

二次方程的一般形式为ax² + bx + c = 0。求解方法包括因式分解法、配方法和公式法。

If the quadratic factorises easily, set each factor equal to zero. For example, x² – 5x + 6 = 0 factors as (x – 2)(x – 3) = 0, so x = 2 or x = 3.

若二次式容易分解，则令每个因式等于0。例如，x² – 5x + 6 = 0分解为(x – 2)(x – 3) = 0，所以x = 2或x = 3。

When factorising is difficult, use the quadratic formula:

当分解困难时，使用求根公式：

x = (-b ± √(b² – 4ac)) / 2a

The value b² – 4ac is called the discriminant. If it is positive, there are two distinct real roots; if zero, one repeated root; if negative, no real roots.

b² – 4ac的值称为判别式。若其大于0，则有两个不等实根；等于0，则有一个重根；小于0，则无实根。

6. Rearranging Formulae | 公式变形

Rearranging a formula means making a different variable the subject. The method is the same as solving an equation: apply inverse operations in the correct order.

公式变形是指把公式中的某个变量作为未知数表示出来，方法同解方程：按正确顺序使用逆运算。

Make r the subject of A = 4πr².

在A = 4πr²中，用A表示r。

Divide both sides by 4π: r² = A/(4π). Then take the positive square root: r = √(A/(4π)) for a radius.

两边同时除以4π：r² = A/(4π)。然后开平方取正根：r = √(A/(4π))（半径取正值）。

If the variable appears in more than one term, collect those terms together and factorise the variable out before dividing.

如果目标变量在多个项中出现，先把含该变量的项合并提取公因式，再两边除以系数。

7. Inequalities | 不等式

Inequalities such as x < 3 or x ≥ -1 are solved in a similar way to equations. The main difference is that multiplying or dividing both sides by a negative number reverses the inequality sign.

像x < 3或x ≥ -1这样的不等式，解法与方程类似。主要区别是：两边同时乘以或除以负数时，不等号方向要改变。

Solve 4 – 2x < 10.

解不等式4 – 2x < 10。

Subtract 4 from both sides: -2x < 6. Divide both sides by -2 and reverse the sign: x > -3.

两边同时减4：-2x < 6。两边同时除以-2并改变不等号方向：x > -3。

You should also be able to show the solution on a number line. Use an open circle for strict inequalities < or >, and a closed circle for ≤ or ≥.

你还应能在数轴上表示解集。对于严格不等式 < 或 > 用空心圆；对于≤或≥用实心圆。

8. Algebraic Fractions | 代数分数

Algebraic fractions work like numerical fractions. To add or subtract them, first rewrite them with a common denominator.

代数分数的运算与分数类似。加减时，先化为同分母。

Simplify 3/(2x) + x/4.

化简3/(2x) + x/4。

The common denominator is 4x. Multiply the first fraction by 2/2 and the second by x/x: 6/(4x) + x²/(4x) = (x² + 6)/(4x).

公分母为4x。第一个分数乘2/2，第二个分数乘x/x：6/(4x) + x²/(4x) = (x² + 6)/(4x)。

When multiplying algebraic fractions, multiply numerators together and denominators together. When dividing, multiply by the reciprocal of the second fraction.

代数分数相乘时，分子乘分子、分母乘分母；相除时，乘以第二个分数的倒数。

9. Indices and Standard Form | 指数与科学计数法

The index laws are essential in algebra. For example: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. Also a⁰ = 1 and a⁻ⁿ = 1/aⁿ.

指数法则是代数中的基础。例如：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，以及(aᵐ)ⁿ = aᵐⁿ。同时a⁰ = 1，a⁻ⁿ = 1/aⁿ。

Standard form expresses numbers as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. It is often used for very large or very small quantities.

科学计数法将数表示为A × 10ⁿ，其中1 ≤ A < 10，n为整数。它常用于表示非常大或非常小的量。

45000 = 4.5 × 10⁴, and 0.00032 = 3.2 × 10⁻⁴

45000 = 4.5 × 10⁴，0.00032 = 3.2 × 10⁻⁴

Be careful with fractional indices: a^(1/2) = √a and a^(1/3) = ∛a.

注意分数指数：a^(1/2) = √a，a^(1/3) = ∛a。

10. Solving Word Problems with Equations | 用方程解决实际问题

Word problems in algebra require translating written statements into equations. Define the unknown with a variable, then set up relationships according to the information given.

代数应用题需要将文字信息转化为方程。先用一个变量表示未知数，再根据题意建立数量关系。

Example: The sum of three consecutive integers is 42. Let the integers be n, n + 1 and n + 2. Then n + (n + 1) + (n + 2) = 42, so 3n + 3 = 42, giving n = 13. The integers are 13, 14 and 15.

例如：三个连续整数的和为42。设它们为n、n + 1和n + 2。则n + (n + 1) + (n + 2) = 42，即3n + 3 = 42，解得n = 13。三个整数为13、14、15。

After solving, always check whether your answer makes sense in the original context. For example, a length or a count must be positive.

解出答案后，务必检查该结果在原始情境中是否合理。例如，长度或数量必须为正。

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IGCSE Mathematics: Core Exam Points in Equations and Algebra | IGCSE数学：方程与代数核心考点梳理