Exercise A: Algebraic Problem-Solving Practice | 练习A：代数解题训练

📚 Exercise A: Algebraic Problem-Solving Practice | 练习A：代数解题训练

This revision exercise is designed to strengthen your algebraic manipulation skills, a cornerstone of the IB Mathematics curriculum. Whether you are following the Analysis and Approaches (AA) or Applications and Interpretation (AI) course, a solid command of algebraic techniques is essential for tackling functions, calculus, and modelling problems with confidence. Exercise A takes you through key topics step by step, from expansion and factorisation to solving equations and inequalities, blending conceptual explanation with worked examples.

本复习练习旨在强化你的代数运算能力，这是 IB 数学课程的核心基础。无论你修读的是分析与方法（AA）还是应用与解释（AI）课程，扎实掌握代数技巧都是攻克函数、微积分和建模问题的关键。练习 A 将带你逐步走遍展开、因式分解、方程与不等式求解等关键主题，并结合概念解析与示范例题，帮助你建立自信。

1. Understanding the Basics of Algebraic Expressions | 理解代数表达式的基础

An algebraic expression contains variables (such as x, y, n), constants, and operational symbols. Terms are separated by addition or subtraction, and each term consists of a coefficient and a variable part. For example, in 3x² + 2x − 5, the coefficient of x² is 3, the coefficient of x is 2, and −5 is the constant term. Recognising like terms is the first step towards simplification.

代数表达式由变量（如 x, y, n）、常数和运算符号组成。项由加法或减法分隔，每一项包含系数和变量部分。例如，在 3x² + 2x − 5 中，x² 的系数是 3，x 的系数是 2，−5 是常数项。识别同类项是进行化简的第一步。

2. Expanding Brackets | 展开括号

xpansion. Special products such as (a + b)² = a² + 2ab + b² and (a − b)(a + b) = a² − b² speed up work in many IB problems.

展开是利用分配律 a(b + c) = ab + ac 去除分组符号的过程。当两个二项式相乘时，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。典型的展开如 (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6。展开后一定要合并同类项。平方差公式和完全平方公式如 (a + b)² = a² + 2ab + b² 和 (a − b)(a + b) = a² − b² 在许多 IB 题中能大幅提升解题速度。

3. Factorisation Techniques | 因式分解技巧

(x − 5)(x + 5). When factorising, always check your result by expanding mentally or on paper.

因式分解是展开的逆过程。首先寻找公因子，如 6x³ − 9x² = 3x²(2x − 3)。对于形如 ax² + bx + c 的二次表达式，尝试写成 (px + q)(rx + s) 的形式。平方差公式经常出现：x² − 25 = (x − 5)(x + 5)。因式分解后务必通过心算或笔算展开来检验结果。

4. Solving Linear Equations | 解一元一次方程

−4. Always substitute your solution back into the original equation to verify correctness.

一元一次方程仅包含一次方的变量，目标是利用逆运算将变量孤立到等式一侧。例如 2(x − 3) = 4x + 2，先展开得 2x − 6 = 4x + 2，接着移项合并：−6 − 2 = 4x − 2x，得出 −8 = 2x，因此 x = −4。任何时候都要把解代回原方程以验证正确性。

5. Solving Quadratic Equations | 解二次方程

solutions are x = 2 or x = 3. When factorisation is not straightforward, use the formula:

x = (−b ± √b² − 4ac) / (2a)

, and Δ < 0 implies no real roots. In IB questions, you may need to express answers in exact simplified surd form.

二次方程可用因式分解、配方法或求根公式求解。若方程 x² − 5x + 6 = 0 可分解为 (x − 2)(x − 3) = 0，则解为 x = 2 或 x = 3。因式分解不便时，使用求根公式：

x = (−b ± √b² − 4ac) / (2a)

记住判别式 Δ = b² − 4ac 揭示根的性质：Δ > 0 有两个相异实根，Δ = 0 有一个重根，Δ < 0 无实数根。在 IB 题目中，你可能需要将答案化为精确的简化根式形式。

6. Manipulating Algebraic Fractions | 代数分式运算

x ≠ 2 and x ≠ −1. When adding or subtracting fractions, find a common denominator, combine numerators, and simplify. Always state restrictions on the variable to avoid division by zero.

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化简代数分式需要先对分子和分母进行因式分解，然后约去公因子。例如 (x² − 4)/(x² − x − 2) 化为 ((x − 2)(x + 2))/((x − 2)(x + 1)) = (x + 2)/(x + 1)，其中 x ≠ 2 且 x ≠ −1。加减分式时，先通分，再合并分子并化简。必须说明变量的限制条件，避免分母为零。

7. Solving Inequalities | 解不等式

ch as x² − x − 6 > 0, solve the associated equation to find critical values x = −2 and x = 3, then test intervals. The solution is x < −2 or x > 3. Use number lines and interval notation to present answers clearly.

解不等式与解方程类似，但有一个关键区别：乘以或除以负数时不等号方向要改变。对于线性不等式 −2x + 5 ≤ 11，先减 5 得 −2x ≤ 6，然后除以 −2 并反转不等号：x ≥ −3。对于二次不等式如 x² − x − 6 > 0，先解出对应的方程得到临界值 x = −2 和 x = 3，然后测试区间。解为 x < −2 或 x > 3。使用数轴和区间符号可以清晰地呈现答案。

8. Functions and Function Notation | 函数与函数符号

of possible outputs. For f(x) = 2x + 1, domain and range are all real numbers. For g(x) = √x − 4, the domain is x ≥ 4 because the square root requires a non‑ adicand. IB questions often ask you to sketch graphs and find values such as f(3) or g(a).

函数 f 将每一个输入 x 对应到唯一的输出 f(x)。定义域是所有允许输入的集合，值域是所有可能输出的集合。对于 f(x) = 2x + 1，定义域和值域均为全体实数。对于 g(x) = √x − 4，由于平方根要求被开方数非负，定义域为 x ≥ 4。IB 题目常要求绘制函数图像，并求 f(3) 或 g(a) 等值。

9. Composite and Inverse Functions | 复合函数与反函数

A composite function f∘g means applying g first, then f: (f∘ e range of f. Always verify that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for values in the domain.

复合函数 f∘g 表示先应用 g，再应用 f：(f∘g)(x) = f(g(x))。要求反函数 f⁻¹，需在 y = f(x) 中交换 x 与 y 并解出 y。对 f(x) = (2x + 1)/3，令 y = (2x + 1)/3，交换得 x = (2y + 1)/3，解得 y = (3x − 1)/2。f⁻¹ 的定义域等于 f 的值域。始终要在定义域内验证 f(f⁻¹(x)) = x 和 f⁻¹(f(x)) = x。

10. Worked Examples Spanning Multiple Skills | 综合技能例题

2 = 3x, so 3x = 3(x² + x − 2). This gives 3x = 3x² + 3x − 6 → 0 = 3x² − 6 → x² = 2 → x = ±√2. Both solutions are valid as they do not violate the restrictions.

我们来解方程 1/(x − 1) + 2/(x + 2) = 3，并说明限制条件。分母不可为零，故 x ≠ 1 且 x ≠ −2。两边同乘以公分母 (x − 1)(x + 2)，得 (x + 2) + 2(x − 1) = 3(x − 1)(x + 2)。化简左边：x + 2 + 2x − 2 = 3x，于是 3x = 3(x² + x − 2)。整理得 3x = 3x² + 3x − 6 → 0 = 3x² − 6 → x² = 2 → x = ±√2。两个解均不违反限制条件，因而有效。

11. Common Mistakes to Avoid | 常见错误避免

Mistakes often arise from sign errors during expansion, forgetting to flip the inequality sign when multiplying by a negative, or cancelling terms instead of factors in fractions. Another frequent slip is misapplying the square root symbol: √x² is |x|, not x. Also, when solving quadratics by factorisation, always bring all terms to one side first; do not divide by the variable hastily because you might lose a solution. In composite functions, apply the order of operations correctly, and check domain restrictions for the inner function.

常见错误包括展开时的符号错误、乘以负数时忘记反转不等号、在分式中约去项而非因子等。另一个易错点是误用平方根符号：√x² 等于 |x|，而非 x。此外，用因式分解解二次方程时，务必先把所有项移到等式一侧，不要贸然除以变量，否则可能丢失解。处理复合函数时，正确执行运算顺序，并留意内层函数的定义域限制。

12. Key Takeaways and Exam Tips | 关键要点与考试技巧

Maintain a systematic approach: simplify, expand, or factorise before jumping into solving. Write down restrictions for fractions and radicals. Use exact values unless specified otherwise. For every answer, perform a quick mental check or substitution. Familiarise yourself with the language of IB command terms such as "hence", "show that", and "determine", as they guide the required working. Finally, practice Exercise A regularly to build fluency, because algebraic agility saves precious time during assessments.

保持系统化的解题思路：先化简、展开或因式分解，再着手求解。记得写出分式和根式的限制条件。除非题目另有要求，一律使用精确值作答。每得出一个答案都快速进行心算检验或代回验证。熟悉 IB 指令术语，如“hence”、“show that”和“determine”，它们引导了所需的解题步骤。最后，定期练习本练习 A 以提升熟练度，因为代数敏捷度能在考评中为你省下宝贵的时间。

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