Quadratic Inequalities | 二次不等式

📚 Quadratic Inequalities | 二次不等式

Quadratic inequalities are a key part of the Edexcel A-Level Pure Mathematics specification. They require you to find the set of x-values for which a quadratic expression is positive, negative, zero or non-negative. Success depends on accurate factorisation, a clear sketch of the parabola and correct use of interval or set notation.

二次不等式是 Edexcel A-Level 纯数学考试大纲的核心内容。题目要求找出使二次函数值为正、负、零或非负的所有 x 值。解题的关键在于准确的因式分解、清晰的抛物线草图，以及正确使用区间或集合符号。

1. What is a quadratic inequality? | 什么是二次不等式

A quadratic inequality is an inequality involving a quadratic expression ax² + bx + c, where a ≠ 0. Typical forms include ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0 and ax² + bx + c ≤ 0.

二次不等式是指含有二次式 ax² + bx + c（a ≠ 0）的不等式。常见形式包括 ax² + bx + c > 0、ax² + bx + c < 0、ax² + bx + c ≥ 0 和 ax² + bx + c ≤ 0。

Unlike a quadratic equation, which usually has at most two solutions, a quadratic inequality usually has infinitely many solutions forming one or two intervals. The graph of y = ax² + bx + c is a parabola, and the inequality asks where this parabola lies above or below the x-axis.

与通常最多只有两个解的二次方程不同，二次不等式通常有无穷多个解，解集为一个或两个区间。函数 y = ax² + bx + c 的图像是一条抛物线，不等式实际上是在寻找这条抛物线位于 x 轴上方或下方的位置。

For Edexcel examinations, you may be asked to give the solution as an inequality, as a set written using set-builder notation, or as an interval using brackets. All three forms must be precise and logically consistent.

在 Edexcel 考试中，题目可能要求将解表示为不等式、使用集合构造符号表示的集合，或使用括号表示的区间。三种形式都必须精确且逻辑一致。

2. Critical values from factorisation | 因式分解求临界值

To solve a quadratic inequality, first solve the related quadratic equation ax² + bx + c = 0. The roots are called critical values because they are the only places where the quadratic expression can change sign.

解二次不等式时，首先应解对应的二次方程 ax² + bx + c = 0。这些根称为临界值，因为它们是二次式可能改变符号的唯一位置。

For example, x² − 5x + 6 factorises as (x − 2)(x − 3), so the critical values are x = 2 and x = 3. These values split the number line into three intervals: x < 2, 2 < x < 3 and x > 3.

例如，x² − 5x + 6 可因式分解为 (x − 2)(x − 3)，因此临界值为 x = 2 和 x = 3。这两个值将数轴分成三个区间：x < 2、2 < x < 3 和 x > 3。

x² − 5x + 6 = (x − 2)(x − 3)

Always factorise fully where possible. If the quadratic does not factorise neatly, use the quadratic formula

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