Algebra: Manipulation and Equations | 代数：运算与方程

📚 Algebra: Manipulation and Equations | 代数：运算与方程

In the Edexcel IGCSE Mathematics syllabus, algebra forms a major part of the assessment. This article provides a concise but thorough revision guide covering algebraic manipulation, equations, inequalities, and key techniques that appear repeatedly in Paper 1 and Paper 2.

在 Edexcel IGCSE 数学大纲中，代数是考试评估的主要部分。本文提供一份简洁而全面的复习指南，涵盖代数运算、方程、不等式，以及在 Paper 1 和 Paper 2 中反复出现的关键技巧。

1. Expanding Brackets | 展开括号

Expanding means removing brackets by multiplying every term inside the bracket by the term outside. For two binomials, use the FOIL method or the distributive law twice.

展开意味着去掉括号，用括号外的每一项乘以括号内的每一项。对于两个二项式，可以使用 FOIL 方法或两次应用分配律。

Example: (x + 3)(x − 2)

示例：(x + 3)(x − 2)

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

Always simplify like terms after expanding.

展开后一定要合并同类项。

2. Factorising | 因式分解

Factorising is the reverse of expanding. Look for common factors first, then special forms: difference of two squares, and quadratics.

因式分解是展开的逆过程。首先寻找公因数，然后识别特殊形式：平方差公式和二次三项式。

a² − b² = (a + b)(a − b)

For a quadratic ax² + bx + c, find two numbers that multiply to ac and add to b. Then split the middle term.

对于二次三项式 ax² + bx + c，找到两个数，它们的乘积等于 ac，和等于 b，然后拆分中间项。

Example: x² + 5x + 6 = (x + 2)(x + 3)

示例：x² + 5x + 6 = (x + 2)(x + 3)

3. Solving Linear Equations | 解线性方程

Linear equations have the unknown only to the first power. Isolate the variable by performing the same operation on both sides.

线性方程中未知数只出现一次幂。通过在两边进行相同的运算来隔离未知数。

Solve: 3x + 5 = 2x − 7

3x − 2x = −7 − 5 ⇒ x = −12

If the unknown appears on both sides, first collect all terms with the unknown on one side and constants on the other.

如果未知数出现在两边，先把含未知数的项移到一边，常数移到另一边。

4. Solving Quadratic Equations | 解二次方程

Quadratic equations can be solved by factorising, completing the square, or using the quadratic formula. Factorisation is often fastest when the roots are rational.

二次方程可以通过因式分解、配方法或求根公式来解。当根为有理数时，因式分解通常最快。

ax² + bx + c = 0 ⇒ x = (−b ± √(b² − 4ac)) / (2a)

Remember the discriminant Δ = b² − 4ac determines the nature of roots.

记住判别式 Δ = b² − 4ac 决定根的性质。

If Δ > 0: two real distinct roots
如果 Δ > 0：两个不等实根
If Δ = 0: one real repeated root
如果 Δ = 0：两个相等实根
If Δ < 0: no real roots
如果 Δ < 0：无实根

5. Simultaneous Equations | 联立方程组

For two linear equations, use substitution or elimination. For a linear and a quadratic equation, substitute the linear expression into the quadratic.

对于两个线性方程，使用代入法或消元法。对于一个线性方程和一个二次方程，将线性表达式代入二次方程中。

Example: y = 2x + 1 and x² + y² = 10

示例：y = 2x + 1 和 x² + y² = 10

x² + (2x + 1)² = 10 ⇒ 5x² + 4x − 9 = 0

Solve the resulting quadratic and then find the corresponding y values.

解所得二次方程，然后求出对应的 y 值。

6. Inequalities | 不等式

Linear inequalities are solved like equations, but if you multiply or divide by a negative number, reverse the inequality sign.

线性不等式的解法与方程类似，但如果乘以或除以一个负数，必须改变不等号方向。

−2x < 6 ⇒ x > −3

For quadratic inequalities, sketch the graph or use a sign table.

对于二次不等式，可以画图或用符号表。

Example: x² − x − 6 < 0

示例：x² − x − 6 < 0

(x − 3)(x + 2) < 0 ⇒ −2 < x < 3

7. Changing the Subject of a Formula | 变换公式主项

Rearrange a formula so that a required variable stands alone on one side. Use inverse operations in reverse order.

重新排列公式，使所需变量单独出现在一边。按逆序使用逆运算。

Example: Make r the subject of V = πr²h

示例：在 V = πr²h 中，用 r 表示公式

r = √(V / (πh))

Be careful with signs and roots when the variable appears in a denominator.

当变量出现在分母中时，注意符号和根号。

8. Indices and Surds | 指数与根式

The laws of indices are essential. For example:

指数法则非常重要。例如：

aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ

Negative and fractional indices also appear in IGCSE:

负指数和分数指数也会出现在 IGCSE 中：

a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a, a^(m/n) = (ⁿ√a)ᵐ

Surds such as √12 simplify to 2√3. Rationalise the denominator when required.

像 √12 这样的根式可以化简为 2√3。必要时分母有理化。

9. Algebraic Fractions | 代数分式

Algebraic fractions follow the same rules as numerical fractions. To add or subtract, use a common denominator.

代数分式与数分式的规则相同。加减时需要通分。

1/(x + 1) + 2/(x − 1) = (x − 1 + 2x + 2) / ((x + 1)(x − 1)) = (3x + 1) / (x² − 1)

Always check for excluded values that make the denominator zero.

始终检查使分母为零的排除值。

10. Sequences and Functions | 数列与函数

Linear sequences have a constant difference; quadratic sequences have a constant second difference. The nth term of a linear sequence is of the form an + b.

线性数列有恒定公差；二次数列有恒定的二阶公差。线性数列的第 n 项形式为 an + b。

Function notation f(x) is used to describe rules. Evaluate by substituting values, and find inverse functions by reversing operations.

函数记号 f(x) 用于描述规则。通过代入数值求值，通过逆运算求反函数。

If f(x) = 2x + 3, then f⁻¹(x) = (x − 3)/2

Composite functions are applied right to left, e.g. fg(x) means g first, then f.

复合函数从右向左运算，例如 fg(x) 表示先算 g，再算 f。

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Algebra: Manipulation and Equations | 代数：运算与方程