IGCSE Cambridge Further Mathematics: Case Study Practical Exercises | IGCSE Cambridge 进阶数学：案例分析实战演练

📚 IGCSE Cambridge Further Mathematics: Case Study Practical Exercises | IGCSE Cambridge 进阶数学：案例分析实战演练

This article presents a series of case-study style worked examples designed to mirror the demands of the Cambridge IGCSE Further Mathematics examination. Each case study focuses on a core topic, showing the full reasoning, the exact techniques, and the exam-ready presentation expected of top-scoring candidates.

本文精选一系列案例分析型实战例题，完全对标剑桥 IGCSE 进阶数学考试风格。每道案例聚焦一个核心考点，完整呈现解题思路、规范步骤和高分答题技巧，帮助考生在考场中做到”见题即破”。

1. The Discriminant and Quadratic Inequalities | 案例一：判别式与二次不等式

Problem. Find the range of values of k for which the equation x² + kx + (k + 3) = 0 has real roots.

题目。求使方程 x² + kx + (k + 3) = 0 有实数根时 k 的取值范围。

For a quadratic equation ax² + bx + c = 0, real roots occur when the discriminant b² − 4ac ≥ 0. Here a = 1, b = k, c = k + 3.

对于一元二次方程 ax² + bx + c = 0，当判别式 b² − 4ac ≥ 0 时方程有实数根。本题中 a = 1，b = k，c = k + 3。

k² − 4(1)(k + 3) ≥ 0 → k² − 4k − 12 ≥ 0 → (k − 6)(k + 2) ≥ 0

Solving the quadratic inequality requires testing the intervals defined by the critical points k = −2 and k = 6. The product is non-negative when k ≤ −2 or k ≥ 6.

解该二次不等式时，以临界点 k = −2 和 k = 6 划分区间进行符号测试。积为非负数时，k ≤ −2 或 k ≥ 6。

Answer: k ≤ −2 or k ≥ 6. In interval notation: (−∞, −2] ∪ [6, ∞).

答案：k ≤ −2 或 k ≥ 6，即 (−∞, −2] ∪ [6, ∞)。

2. Logarithmic Equations Using Change of Base | 案例二：对数换底公式的应用

Problem. Solve log₂x + logₓ8 = 4 for

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