二次不等式求解全攻略：从入门到精通 | Solving Quadratic Inequalities: A Complete Guide

📐 什么是二次不等式？| What Are Quadratic Inequalities?

二次不等式是形如 ax² + bx + c > 0（或 <, ≥, ≤）的不等式，是GCSE Higher Tier和A Level数学的核心考点。掌握它不仅能帮你拿下考试中的高频题型，更是理解函数图像与代数关系的桥梁。

A quadratic inequality takes the form ax² + bx + c > 0 (or <, ≥, ≤). It’s a core topic in both GCSE Higher Tier and A Level Mathematics. Mastering it not only secures high-frequency exam marks but also bridges algebraic reasoning with graphical intuition.

🔑 五大核心知识点 | 5 Key Concepts

1. 标准化变形 | Rearrange to Standard Form

第一步永远是把不等式整理成 ax² + bx + c > 0 的标准形式，所有项移到左边，右边归零。

Always start by rearranging into the standard form ax² + bx + c > 0 — move everything to the left, zero on the right.

2. 因式分解求临界值 | Factorise to Find Critical Values

解对应的二次方程 ax² + bx + c = 0，通过因式分解找出 x 轴截距（critical values）。例如：x² + 2x − 8 = 0 → (x+4)(x−2) = 0 → x = −4, x = 2。

Solve the associated quadratic equation ax² + bx + c = 0 by factorising to find the x-intercepts. E.g.: x² + 2x − 8 = 0 → (x+4)(x−2) = 0 → x = −4, x = 2.

3. 画草图定位区间 | Sketch the Parabola

根据 a 的正负画出抛物线的开口方向，标注 x 截距和 y 截距。a > 0 开口向上（∪），a < 0 开口向下（∩）。

Sketch the parabola based on the sign of a: a > 0 opens upward (∪), a < 0 opens downward (∩). Mark the x-intercepts and y-intercept clearly.

4. 确定满足条件的区域 | Identify the Satisfying Region

不等式要求 > 0 时取 x 轴上方区域，< 0 时取 x 轴下方区域。例如 x² + 2x − 8 ≥ 0 → x ≤ −4 或 x ≥ 2。

For > 0, take regions above the x-axis; for < 0, take regions below. E.g., x² + 2x − 8 ≥ 0 → x ≤ −4 or x ≥ 2.

5. 数轴表示答案 | Present on a Number Line

用实心圆点（≥/≤）或空心圆点（>/<）表示临界值，画箭头表示解集区间。考试中数轴图示往往是得分关键！

Use solid dots for ≥/≤ and open dots for >/<. Draw arrows to represent the solution intervals. A clear number line diagram often earns you those final marks!

📚 学习建议 | Study Tips

• 先判断开口方向 — a 的正负决定了不等号方向的含义 | Check the parabola direction first — the sign of a determines the meaning of the inequality.
• 熟记口诀：大于取两边，小于取中间（当 a > 0 时）| Mnemonic: “Greater → outside, Less → between” (when a > 0).
• 多做历年真题 — Past Papers 是最有效的训练方式 | Practice past papers — nothing beats real exam questions for building intuition.

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