📚 Solving Quadratic Inequalities | 求解二次不等式
Quadratic inequalities appear frequently in IGCSE Edexcel Mathematics, often asking you to find the range of x-values that satisfy an expression like ax² + bx + c > 0. While solving quadratic equations gives you specific points, inequalities describe entire intervals on the number line. Mastering this topic requires a solid understanding of factorising, sketching parabolas, and interpreting where a quadratic graph lies above or below the x-axis. This article covers all the essential methods, from basic factorable quadratics to cases with negative leading coefficients and no real roots.
二次不等式在IGCSE Edexcel数学考试中经常出现，通常要求找出使 ax² + bx + c > 0 成立的 x 的取值范围。解二次方程给出的是具体的点，而不等式描述的是数轴上的整个区间。掌握这一部分需要扎实的因式分解基础、能画出抛物线草图，并会判断二次函数的图像在 x 轴上方还是下方。本文涵盖所有重要方法，从最基本的可因式分解的二次式，到首项系数为负以及无实数根的情形。
1. What is a Quadratic Inequality? | 什么是二次不等式？
a, b, and c are constants and a ≠ 0. Instead of solving for exact values of x, we find a set of x-values, often an interval or a union of intervals, that make the inequality true.
二次不等式是含有二次表达式的不等式，例如 ax² + bx + c < 0、ax² + bx + c ≤ 0、ax² + bx + c > 0 或 ax² + bx + c ≥ 0，其中 a, b, c 是常数且 a ≠ 0。与解方程不同，我们不是求出 x 的精确值，而是找出使不等式成立的一个集合，通常是数轴上的一个区间或多个区间的并集。
2. Standard Form and Direction of Inequality | 标准形式与不等号方向
Always start by rearranging the inequality so that one side is zero, with the quadratic expression on the other side. For example, bring all terms to the left: ax² + bx + c > 0. If you multiply or divide both sides by a negative number, remember to reverse the direction of the inequality sign. This step is crucial when the coefficient of x² is negative.
第一步总是将不等式整理成一边为零、另一边为二次表达式的标准形式，例如 ax² + bx + c > 0。如果两边同时乘以或除以一个负数，必须注意反转不等号的方向。当 x² 的系数为负时，这一步格外关键。
Example: -x² + 3x + 4 > 0 → Multiply by -1: x² - 3x - 4 < 0
示例：-x² + 3x + 4 > 0 → 两边乘以 -1：x² - 3x - 4 < 0
3. The Graphical Method: Thinking in Parabolas | 图像法：用抛物线思考
abola. When a > 0, the parabola opens upwards (U-shaped); when a < 0, it opens downwards (∩ ality ax² + bx + c > 0 correspond to the x-values where the graph is above the x-axis (y > 0). For < 0, we look for where the graph is below the x-axis.
y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上（U 形）；当 a < 0 时，开口向下（∩ 形）。不等式 ax² + bx + c > 0 的解对应的是图像在 x 轴上方的 x 取值范围（y > 0）。对于 < 0 的情形，则是图像在 x 轴下方的部分。
4. Finding Critical Values: Roots of the Quadratic | 求临界值：二次方程的根
s, write it in the form (px + q)(rx + s) = 0. If not, use the quadratic formula: x = [-b ± √b² - 4ac] / (2a). Be careful with the discriminant; we will discuss cases where it is negative later.
先解对应的二次方程 ax² + bx + c = 0，求出图像与 x 轴的交点，这些就是划分区间的临界值。如果能因式分解，写成 (px + q)(rx + s) = 0 的形式。如果不能分解，就用求根公式：x = [-b ± √b² - 4ac] / (2a)。要注意判别式的情况，后面会讨论当判别式为负时的处理方法。
5. Sketching the Parabola and Testing Intervals | 画抛物线草图并检验区间
q (with p < q), draw a quick sketch of the parabola, showing whether it opens up or down. Mark the roots on the x-axis. The parabola will change sign at each root. For a > 0, the graph is below the x-axis between the roots and above the x-axis outside the roots. For a < 0, the pattern is reversed. You can also test a number from each interval in the factored inequality to confirm the sign.
求出两个根之后，假设为 x = p 和 x = q（且 p < q），快速画出抛物线的草图，标明开口方向。在 x 轴上标出这两个根。抛物线每经过一个根就会改变符号。当 a > 0 时，图像在两个根之间位于 x 轴下方，在两根之外位于 x 轴上方。当 a < 0 时，情况相反。你也可以在每个区间内选取一个测试值代入分解后的不等式来验证符号。
6. Worked Example 1: Positive Leading Coefficient | 范例1：首项系数为正
ns upwards. It is above the x-axis when x < 2 or x > 3. Therefore, the solution is x < 2 or x > 3. In set notation: {x : x < 2} ∪ {x : x > 3}.
解 x² - 5x + 6 > 0。因式分解得 (x - 2)(x - 3) > 0。临界值为 x = 2 和 x = 3。因为 a = 1 > 0，抛物线开口向上。在 x < 2 或 x > 3 时，图像位于 x 轴上方。因此解为 x < 2 或 x > 3。用集合表示法为 {x : x < 2} ∪ {x : x > 3}。
7. Worked Example 2: Inclusive Inequality (≤ or ≥) | 范例2：带等号的不等式（≤ 或 ≥）
s including the endpoints. Thus the solution is 0 ≤ x ≤ 4, written as the interval [0, 4].
解 x² - 4x ≤ 0。因式分解得 x(x - 4) ≤ 0。根为 0 和 4。抛物线开口向上。表达式 ≤ 0 对应图像在 x 轴上或 x 轴下方，即两根之间并包括端点。因此解为 0 ≤ x ≤ 4，区间表示为 [0, 4]。
8. Worked Example 3: Negative Leading Coefficient | 范例3：首项系数为负
een the roots: -1 < x < 4. Always remember to reverse the inequality when multiplying by a negative.
解 -x² + 3x + 4 > 0。两边乘以 -1 并反转不等号得 x² - 3x - 4 < 0。因式分解为 (x - 4)(x + 1) < 0。临界值为 x = -1 和 x = 4。此时抛物线开口向上（因 a = 1 > 0）。我们需要 y < 0，即两根之间的部分：-1 < x < 4。切记，当两边同乘负数时一定要反转不等号。
9. When the Quadratic Has No Real Roots | 当二次式无实数根时
upwards, so it is always positive for all real x. The solution is all real numbers. Conversely, x² + x + 1 < 0 has no solution.
如果判别式 b² - 4ac < 0，则二次函数与 x 轴没有交点。整个抛物线要么完全在 x 轴上方，要么完全在下方。例如 x² + x + 1 > 0，判别式 = 1 - 4 = -3 < 0，且抛物线开口向上，因此对所有实数 x 恒为正，解为所有实数。反之，x² + x + 1 < 0 则无解。
Always check the sign of a to decide whether the expression is always positive or always negative.
利用 a 的正负来判断表达式是恒正还是恒负。
10. Expressing Solutions Using Set Notation and Intervals | 用集合符号和区间表示解
clearly. You can use inequality notation (e.g., x < 2 or x > 3), set notation ({x : x < 2} ∪ {x : x > 3}), or interval notation ((-∞, 2) ∪ kets for strict inequalities (<, >) and square brackets for inclusive inequalities (≤, ≥).
IGCSE Edexcel 要求能够清晰地写出解。你可以使用不等式记法（如 x < 2 或 x > 3）、集合记法（{x : x < 2} ∪ {x : x > 3}）或区间记法（(-∞, 2) ∪ (3, ∞)）。在临界值处要注意开闭区间：严格不等式（<, >）用圆括号，带等号的不等式（≤, ≥）用方括号。
11. Common Pitfalls and How to Avoid Them | 常见易错点及避免方法
One common mistake is forgetting to reverse the inequality sign when multiplying or dividing by a negative. Another is incorrectly identifying the 'inside' or 'outside' regions; always sketch the graph or use a sign table to confirm. Students also sometimes solve only the equation and forget that inequalities describe intervals. Finally, when the quadratic does not factorise, ensure you correctly apply the quadratic formula and simplify the surd form if needed.
一个常见错误是两边乘或除以负数时忘记反转不等号。另一个是混淆“中间”和“两边”的区域；一定要画出图像或使用符号表来确认。有的同学只解了方程，而忘了不等式给出的是区间。最后，当二次式不能因式分解时，确保正确使用求根公式，并在需要时化简根式。
12. Summary and Exam Tips | 总结与备考建议
To master quadratic inequalities, always rearrange to standard form, factorise or use the quadratic formula to find critical values, sketch the parabola, and then read off the solution intervals. Remember to check whether the inequality includes equality. Practise with a variety of examples, including those where you must first divide by a negative coefficient. In the exam, show your working clearly and double-check the direction of the inequality in your final answer.
要掌握二次不等式，务必先整理成标准形式，通过因式分解或求根公式找出临界值，画出抛物线草图，然后直接读出解的区间。注意不等式是否包含等号。多做各式各样的练习题，包括需要先处理负系数的题目。考试时，写出清晰的解题步骤，并在最后的答案中再次检查不等号的方向是否正确。
Published by TutorHao | IGCSE Edexcel Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图