📚 Solving Quadratic Inequalities | 解二次不等式
method for solving quadratic inequalities of the form ax² + bx + c > 0, < 0, ≥ 0, or ≤ 0.
二次不等式是 Edexcel IGCSE 数学大纲中的一个核心主题。它不仅考查代数运算能力，还考查对二次函数图像性质的理解。在本文中，我们将系统学习如何求解形如 ax² + bx + c > 0、< 0、≥ 0 或 ≤ 0 的二次不等式。
1. What is a Quadratic Inequality? | 什么是二次不等式？
eater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤).
二次不等式是含有二次表达式的不等式。其一般形式是将 ax² + bx + c 与零进行比较，其中 a ≠ 0。比较关系可以是大于（>）、小于（<）、大于或等于（≥）、小于或等于（≤）。
For example, x² − 4x + 3 > 0 and 2x² + 5x − 3 ≤ 0 are both quadratic inequalities.
例如，x² − 4x + 3 > 0 和 2x² + 5x − 3 ≤ 0 都是二次不等式。
The key difference between a quadratic equation and a quadratic inequality is that an equation gives specific solutions (roots), while an inequality gives a range of values, often expressed as intervals of x.
二次方程与二次不等式之间的关键区别在于：方程给出确定的解（根），而不等式则给出一个取值范围，通常以 x 的区间形式表达。
2. The Quadratic Graph Approach | 二次函数图像法
To understand quadratic inequalities deeply, we must first recall the shape of a quadratic graph. When a > 0, the parabola opens upwards (U-shaped); when a < 0, it opens downwards (∩-shaped).
要深入理解二次不等式，我们必须首先回顾二次函数图像的形状。当 a > 0 时，抛物线开口朝上（U 形）；当 a < 0 时，开口朝下（∩ 形）。
. The roots are x = 5 and x = −2. These are the points where the curve crosses the x-axis.
考虑函数 f(x) = x² − 3x − 10。因式分解得 f(x) = (x − 5)(x + 2)。根为 x = 5 和 x = −2。这些是曲线与 x 轴相交的点。
Between the roots, the curve lies below the x-axis (f(x) < 0). Outside the roots, the curve lies above the x-axis (f(x) > 0). This simple observation is the foundation of our solution strategy.
在两根之间，曲线位于 x 轴下方（f(x) < 0）。在两根之外，曲线位于 x 轴上方（f(x) > 0）。这一简单观察就是我们解题策略的基础。
3. Finding the Critical Values | 求关键值（临界值）
The critical values are the roots of the corresponding quadratic equation. To find them, we set the quadratic expression equal to zero and solve:
关键值就是对应二次方程的根。要求它们，我们将二次表达式设为零并求解：
ax² + bx + c = 0
This is achieved by factorising, completing the square, or using the quadratic formula:
这可以通过因式分解、配方法或求根公式来实现：
x = (−b ± √b² − 4ac) ⁄ (2a)
For the Edexcel IGCSE exam, factorising is the most common and preferred method, as most exam questions are designed with integer roots.
对于 Edexcel IGCSE 考试而言，因式分解是最常用且首选的方法，因为大多数考题都设计为整数根。
Worked Example: Solve x² − 3x − 10 = 0 to find critical values.
示例：解方程 x² − 3x − 10 = 0 以求出关键值。
Factorise: (x − 5)(x + 2) = 0. Therefore x = 5 or x = −2.
因式分解：(x − 5)(x + 2) = 0。因此 x = 5 或 x = −2。
4. Solving x² − 3x − 10 > 0 | 求解 x² − 3x − 10 > 0
Now let us solve the inequality x² − 3x − 10 > 0 step by step.
现在我们逐步求解不等式 x² − 3x − 10 > 0。
Step 1: Factorise the quadratic expression.
步骤 1：对二次表达式进行因式分解。
x² − 3x − 10 = (x − 5)(x + 2)
Step 2: Find the critical values by setting each factor to zero: x = 5 and x = −2.
步骤 2：令每个因式为零，求关键值：x = 5 和 x = −2。
Step 3: Mark these two values on a number line. They divide the line into three regions:
步骤 3：在数轴上标出这两个值。它们将数轴分为三个区域：
- Region A: x < −2
- Region B: −2 < x < 5
- Region C: x > 5
Step 4: Test a value from each region in the original inequality.
步骤 4：从每个区域取一个值代入原不等式进行检验。
- Region A: try x = −3. Then (−3)² − 3(−3) − 10 = 9 + 9 − 10 = 8 > 0 ✓
- Region B: try x = 0. Then 0² − 0 − 10 = −10 < 0 ✗
- Region C: try x = 6. Then 6² − 18 − 10 = 36 − 18 − 10 = 8 > 0 ✓
Step 5: The solution is the union of the regions where the inequality is satisfied:
步骤 5：解为满足不等式的区域的并集：
x < −2 or x > 5
In interval notation: x ∈ (−∞, −2) ∪ (5, ∞).
用区间表示：x ∈ (−∞, −2) ∪ (5, ∞)。
5. Solving x² − 3x − 10 < 0 | 求解 x² − 3x − 10 < 0
Using the same critical values, we now solve the opposite inequality x² − 3x − 10 < 0.
使用相同的关键值，我们现在求解相反的不等式 x² − 3x − 10 < 0。
From our region testing above, we saw that the expression is negative only in Region B (−2 < x < 5).
从上述区域检验中，我们看到表达式仅在区域 B（−2 < x < 5）中为负。
Therefore the solution is:
因此解为：
−2 < x < 5
This corresponds to the interval x ∈ (−2, 5).
这对应于区间 x ∈ (−2, 5)。
Key Insight: For a positive quadratic (a > 0): - The inequality > 0 (or ≥ 0) gives solutions "outside" the roots. - The inequality < 0 (or ≤ 0) gives solutions "between" the roots.
关键要点：对于正系数二次函数（a > 0）： - 不等式 > 0（或 ≥ 0）的解在两根之外。 - 不等式 < 0（或 ≤ 0）的解在两根之间。
Remember the mnemonic: "First outside, then between" — or think of the U-shaped smile of the parabola: the smile is above the x-axis at the edges.
记住口诀：「外大于、内小于」——或者想象抛物线 U 形的微笑：微笑的两端在 x 轴上方。
6. The Sign Diagram Method | 符号图表法
The sign diagram is a more formal and efficient alternative to region testing. It is particularly useful in the exam because it is quick and less prone to arithmetic errors.
符号图表是区域检验法的一种更正式且更高效的方法。它在考试中特别有用，因为速度快且不容易出现计算错误。
Step 1: Factorise and identify critical values.
步骤 1：因式分解并找出关键值。
Step 2: List the critical values in increasing order on a number line.
步骤 2：在数轴上按递增顺序列出关键值。
Step 3: A linear factor (x − k) is negative when x < k, zero when x = k, and positive when x > k.
步骤 3：对每个因式，判断其在每个区域的符号。线性因式 (x − k) 在 x < k 时为负，在 x = k 时为零，在 x > k 时为正。
Step 4: Multiply the signs together to determine the sign of the whole expression in each region.
步骤 4：将符号相乘，确定整个表达式在每个区域的符号。
Let us construct a sign diagram for (x − 5)(x + 2):
让我们为 (x − 5)(x + 2) 构建符号图表：
| Region | x < −2 | −2 < x < 5 | x > 5 |
| (x + 2) | Negative (−) | Positive (+) | Positive (+) |
| (x − 5) | Negative (−) | Negative (−) | Positive (+) |
| Product | (+) — above axis | (−) — below axis | (+) — above axis |
This table immediately reveals that the product is positive for x < −2 or x > 5, and negative for −2 < x < 5.
该表立即揭示出：乘积在 x < −2 或 x > 5 时为正，在 −2 < x < 5 时为负。
With practice, you can draw this sign diagram in under a minute, making it ideal for the exam environment.
通过练习，你可以在一分钟内画出这个符号图表，非常适合考试环境。
7. When the Coefficient a < 0 | 当系数 a < 0 时
at happens when the quadratic has a negative leading coefficient, such as −x² + 4x + 5 ≥ 0? We have two options.
当二次函数首项系数为负时，例如 −x² + 4x + 5 ≥ 0，该怎么办？我们有两种选择。
Option 1 — Multiply by −1: Multiply both sides by −1 and reverse the inequality sign. This is usually the cleanest approach.
方法 1 — 乘以 −1：将不等式两边同时乘以 −1，并反转不等号。这通常是最简洁的方法。
−x² + 4x + 5 ≥ 0 ⟹ x² − 4x − 5 ≤ 0
Option 2 — Work with the negative quadratic: The parabola opens downwards, so the graph is above the x-axis between the roots and below the x-axis outside the roots. This reverses the relationship from the a > 0 case.
方法 2 — 直接处理负二次函数：抛物线开口朝下，因此图像在两根之间位于 x 轴上方，在两根之外位于 x 轴下方。这与 a > 0 的情况正好相反。
al values are −1 and 5. Since the inequality is ≤ 0, we take the region between the roots:
我们完成方法 1。因式分解 x² − 4x − 5 = (x − 5)(x + 1) ≤ 0。关键值为 −1 和 5。由于不等式为 ≤ 0，我们取两根之间的区域：
−1 ≤ x ≤ 5
Important: Never forget to reverse the inequality sign when multiplying or dividing by a negative number. This is one of the most common careless errors in IGCSE examinations.
重要：在乘以或除以负数时，切勿忘记反转不等号。这是 IGCSE 考试中最常见的粗心错误之一。
8. Inequalities with ≥ and ≤ | 含 ≥ 和 ≤ 的不等式
lities (≥, ≤) lies in whether the critical values themselves are included in the solution.
严格不等式（>、<）与非严格不等式（≥、≤）之间的区别在于关键值本身是否包含在解中。
d because the expression equals zero at those points, and zero satisfies both ≥ 0 and ≤ 0.
对于 ≥ 0 或 ≤ 0，根被包含在解中，因为在那些点上表达式等于零，而零同时满足 ≥ 0 和 ≤ 0。
lution of x² − 3x − 10 ≥ 0 is x ≤ −2 or x ≥ 5, using square brackets in interval notation:
例如，x² − 3x − 10 ≥ 0 的解为 x ≤ −2 或 x ≥ 5，在区间表示中使用方括号：
x ∈ (−∞, −2] ∪ [5, ∞)
And the solution of x² − 3x − 10 ≤ 0 is −2 ≤ x ≤ 5, or x ∈ [−2, 5].
而 x² − 3x − 10 ≤ 0 的解为 −2 ≤ x ≤ 5，即 x ∈ [−2, 5]。
The graph makes this clear: at the roots, the parabola touches the x-axis, so those points satisfy the equality condition. Use a closed dot ● on the number line for included values, and an open dot ○ for excluded values.
图像清楚地说明了这一点：在根处，抛物线接触 x 轴，因此这些点满足等号条件。数轴上用实心点 ● 表示包含该值，用空心点 ○ 表示不包含该值。
9. Perfect Squares and No Real Roots | 完全平方与无实数根
Not all quadratic inequalities factorise into two distinct linear factors. Two special cases require extra attention.
并非所有二次不等式都能分解为两个不同的线性因式。两种特殊情况需要特别注意。
Case 1 — Perfect square: Suppose x² − 6x + 9 > 0. This factorises to (x − 3)² > 0.
情形 1 — 完全平方：假设 x² − 6x + 9 > 0。它分解为 (x − 3)² > 0。
only when x = 3. Therefore the solution to (x − 3)² > 0 is all real numbers except x = 3:
完全平方总是大于或等于零。它仅在 x = 3 时等于零。因此 (x − 3)² > 0 的解是所有实数除了 x = 3：
x ∈ ℝ, x ≠ 3
)² ≥ 0, the solution would simply be all real numbers, since the square is never negative.
如果不等式为 (x − 3)² ≥ 0，解就是所有实数，因为平方永远不会为负。
Case 2 — No real roots: 0, the parabola opens upwards and lies entirely above the x-axis. Therefore the inequality is true for all real x.
情形 2 — 无实数根：考虑 x² + 2x + 5 > 0。判别式为 b² − 4ac = 4 − 20 = −16 < 0，因此该二次函数没有实数根。抛物线不与 x 轴相交。由于 a = 1 > 0，抛物线开口朝上且完全位于 x 轴上方。因此该不等式对所有实数 x 均成立。
In contrast, x² + 2x + 5 < 0 would have no solution at all, because a positive quadratic can never be negative if it never crosses the x-axis.
相反，x² + 2x + 5 < 0 将完全没有解，因为一个不穿越 x 轴的正系数二次函数永远不可能为负。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Here are the most frequent errors students make in the Edexcel IGCSE exam when dealing with quadratic inequalities, along with strategies to avoid them.
以下是学生在 Edexcel IGCSE 考试中处理二次不等式时最常见的错误，以及相应的避免策略。
- Forgetting the direction of the parabola. Always check the sign of a. If a < 0, the 'outside-between' rule reverses. Solution: multiply by −1 first.
- 错误一：忘记抛物线的开口方向。始终检查 a 的符号。如果 a < 0，则「外大于、内小于」的规律会反转。解决办法：先将不等式两边乘以 −1。
- Mixing up the inequality sign after multiplying by a negative. Example: −x² + 4 > 0 must become x² − 4 < 0, not still > 0.
- 错误二：乘以负数后没有反转不等号。例如：−x² + 4 > 0 必须变为 x² − 4 < 0，而不是保持不变。
- Writing the answer as two separate intervals without the word 'or'. For > 0 with two roots, write x < a or x > b — never write x < a and x > b as a single statement.
- 错误三：将两个分离区间未用「或」连接。对于 > 0 且有两个根的情况，应写为 x < a 或 x > b——绝不能将其作为单一语句写成 x < a 且 x > b。
- Forgetting to include roots when the sign is ≥ or ≤. Check whether the critical values should use solid dots (included) or hollow dots (excluded).
- 错误四：当符号为 ≥ 或 ≤ 时忘记包含根。检查关键值应使用实心点（包含）还是空心点（不包含）。
- Not testing the regions. Test one value from each region to confirm your algebraic conclusion matches the actual sign pattern.
- 错误五：没有检验各区域。从每个区域取一个值代入检验，以确认代数结论与实际符号模式一致。
11. Worked Exam-Style Examples | 考试风格例题
Let us work through three full examples in the format you will encounter in the Edexcel IGCSE examination.
让我们以 Edexcel IGCSE 考试中会遇到的形式，完成三个完整例题。
Example A: Solve 2x² − 5x − 3 < 0.
例题 A：求解 2x² − 5x − 3 < 0。
Factorise: 2x² − 5x − 3 = (2x + 1)(x − 3). Critical values satisfy 2x + 1 = 0 ⟹ x = −½, and x − 3 = 0 ⟹ x = 3.
因式分解：2x² − 5x − 3 = (2x + 1)(x − 3)。关键值满足 2x + 1 = 0 ⟹ x = −½，以及 x − 3 = 0 ⟹ x = 3。
Since a = 2 > 0, the parabola opens upwards. The inequality is 'less than zero', so we take the region between the roots:
由于 a = 2 > 0，抛物线开口朝上。不等式为「小于零」，因此我们取两根之间的区域：
−½ < x < 3
Example B: Solve x² + 6x + 9 ≤ 0.
例题 B：求解 x² + 6x + 9 ≤ 0。
makes it equal to zero is x = −3. Since it can never be negative, the solution is simply:
因式分解：x² + 6x + 9 = (x + 3)² ≤ 0。完全平方始终非负。唯一使其等于零的值是 x = −3。由于它永远不能为负，解法就是：
x = −3
Example C: Solve −x² + 3x + 4 > 0.
例题 C：求解 −x² + 3x + 4 > 0。
Multiply by −1 and reverse the sign: x² − 3x − 4 < 0. Factorise: (x − 4)(x + 1) < 0. Roots are 4 and −1. The solution lies between the roots:
乘以 −1 并反转符号：x² − 3x − 4 < 0。因式分解：(x − 4)(x + 1) < 0。根为 4 和 −1。解在两根之间：
−1 < x < 4
12. Summary of Key Points | 要点总结
Let us consolidate everything we have learned into a concise revision checklist.
让我们将所学内容浓缩为一份简洁的复习清单。
- Always rearrange the inequality so that one side is zero.
- 总是将不等式整理为一边为零的形式。
- Factorise the quadratic and find the critical values (roots).
- 对二次式进行因式分解并求出关键值（根）。
- If necessary, multiply by −1 to make a > 0, remembering to reverse the inequality sign.
- 如有必要，乘以 −1 使 a > 0，并记得反转不等号。
- Sketch the parabola or construct a sign diagram.
- 画出抛物线草图或构建符号图表。
- For a > 0: > 0 gives x < root₁ or x > root₂; < 0 gives root₁ < x < root₂.
- 对于 a > 0：> 0 给出 x < 根₁ 或 x > 根₂；< 0 给出 根₁ < x < 根₂。
- Use ≥/≤ to include the roots; use >/< to exclude them.
- 使用 ≥/≤ 时包含根；使用 >/< 时排除根。
- Handle special cases: perfect squares touch the axis at one point; positive quadratics with a negative discriminant are always positive.
- 处理特殊情况：完全平方在一点处接触 x 轴；判别式为负的正系数二次函数恒为正。
Mastery of quadratic inequalities comes from practice. Work through every question in your Edexcel IGCSE past papers, and draw a sketch for every problem until the graphical intuition becomes second nature. This skill will serve you well both in Paper 1 and Paper 2.
掌握二次不等式需要大量练习。仔细完成 Edexcel IGCSE 历年真题中的每一道题目，并为每道题画草图，直到图形直觉成为你的第二天性。这一技能在 Paper 1 和 Paper 2 中都将使你受益良多。
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