📚 Quadratic Inequalities | 二次不等式
Quadratic inequalities may look a little intimidating at first, but they follow a clear, logical process that builds on your knowledge of solving quadratic equations and understanding graphs. This article will guide you step by step through the methods needed to tackle any quadratic inequality in the IGCSE Edexcel Mathematics syllabus, from factorising to using the quadratic formula and representing solutions on a number line.
二次不等式初看可能有些吓人，但它的求解过程清晰而有逻辑，完全建立在你已掌握的二次方程求解和图像理解的基础之上。本文将逐步带你掌握IGCSE Edexcel数学大纲中所有与二次不等式相关的方法，从因式分解到使用求根公式，再到在数轴上表示解集。
1. What Are Quadratic Inequalities? | 什么是二次不等式？
A quadratic inequality is an inequality that involves a quadratic expression, such as ax² + bx + c < 0 or ax² + bx + c ≥ 0, where a, b and c are constants and a ≠ 0. Instead of finding specific values for x that make the expression equal to zero, you are looking for a range (or ranges) of x-values that make the inequality true.
二次不等式是指含有二次表达式的 不等式，例如 ax² + bx + c < 0 或 ax² + bx + c ≥ 0，其中 a、b 和 c 为常数且 a ≠ 0。与求解使表达式等于零的具体 x 值不同，你需要找出使不等式成立的一个或多个 x 取值范围。
For example, x² − 4x + 3 > 0 asks: ‘For which x-values does the parabola y = x² − 4x + 3 lie above the x-axis?’ Answering this question merges your algebraic and graphical understanding of quadratics.
例如，x² − 4x + 3 > 0 问的是：“对于哪些 x 值，抛物线 y = x² − 4x + 3 位于 x 轴上方？” 回答这一问题需要你结合对二次函数的代数理解和图像理解。
2. The General Form | 一般形式
All quadratic inequalities can be rearranged into one of four standard forms: ax² + bx + c < 0, ax² + bx + c ≤ 0, ax² + bx + c > 0, or ax² + bx + c ≥ 0 on one side before you begin solving. For instance, 2x² − 5x ≤ 3 becomes 2x² − 5x − 3 ≤ 0.
所有二次不等式都可以整理为四种标准形式之一：ax² + bx + c < 0、ax² + bx + c ≤ 0、ax² + bx + c > 0 或 ax² + bx + c ≥ 0。在开始求解之前，必须将不等式整理为一边是零的形式。例如，2x² − 5x ≤ 3 应整理为 2x² − 5x − 3 ≤ 0。
Having a positive coefficient of x² makes sketching the graph much easier. If a < 0, multiply the whole inequality by −1, remembering to reverse the inequality symbol. Thus, −x² + 2x − 3 > 0 becomes x² − 2x + 3 < 0.
让 x² 的系数为正会使画图变得更加容易。如果 a < 0，则将整个不等式乘以 −1，并记得要反转不等号。因此，−x² + 2x − 3 > 0 变为 x² − 2x + 3 < 0。
3. Step 1 – Solve the Corresponding Quadratic Equation | 步骤一：解对应的二次方程
Treat the inequality as an equation by replacing the inequality sign with an equals sign. Solve ax² + bx + c = 0 using factorisation, completing the square, or the quadratic formula. The solutions are called the critical values, and they divide the number line into distinct intervals.
将不等号替换为等号，把不等式视为方程来解。用因式分解、配方法或求根公式解 ax² + bx + c = 0。所得的解称为临界值，它们将数轴划分为不同的区间。
− 6 = 0, which factorises to (x − 3)(x + 2) = 0. The critical values are x = 3 and x = −2.
例如，考虑不等式 x² − x − 6 > 0。对应的方程为 x² − x − 6 = 0，因式分解得到 (x − 3)(x + 2) = 0。临界值为 x = 3 和 x = −2。
4. Step 2 – Find Critical Values | 步骤二：找出临界值
Critical values are simply the roots of the quadratic equation. If the quadratic does not factorise easily, use the quadratic formula:
临界值就是二次方程的根。如果二次式不易因式分解，可使用求根公式：
x = [−b ± √b² − 4ac] / 2a
Always check the discriminant b² − 4ac first. If it is negative, the quadratic has no real roots, which means the expression is either always positive or always negative. In such cases the inequality has either no solution or all real numbers as solution, depending on the sign of a.
一定先检查判别式 b² − 4ac。如果判别式为负，则二次式没有实数根，这意味着该表达式要么恒为正，要么恒为负。在这种情况下，不等式要么无解，要么解集为全体实数，具体取决于 a 的符号。
A zero discriminant gives one repeated root, meaning the parabola touches the x-axis at a single point. This will affect whether the inequality includes equality at that point.
判别式为零时，得到二重根，即抛物线与 x 轴仅在一个点相切。这将影响不等式是否在该点处包含等号。
5. Step 3 – Sketch the Parabola or Use a Sign Table | 步骤三：画出抛物线或使用符号表
or negative, sketch the graph of y = ax² + bx + c. If a > 0, the parabola opens upwards (∪-shaped); if a < 0, it opens downwards (∩ regions where the curve is above the x-axis give y > 0, and the regions below give y < 0.
要判断表达式在哪些区间为正或为负，可以画出 y = ax² + bx + c 的图像。若 a > 0，抛物线开口向上（∪ 形）；若 a < 0，则开口向下（∩ 形）。标出图像与 x 轴相交的临界值。曲线在 x 轴上方的区域给出 y > 0，下方的区域给出 y < 0。
Alternatively, a sign table can be used by testing a sample x-value from each interval between the critical values. Substitute the test value into the factorised quadratic to see whether the product is positive or negative in that interval.
也可以使用符号表，在临界值之间的每个区间各取一个测试 x 值，将其代入因式分解后的二次式，判断乘积在该区间的正负。
6. Determining the Solution Interval for > 0 | 确定大于零的解区间
When solving ax² + bx + c > 0 with a > 0, the parabola opens upwards. The expression is greater than zero outside the interval between the two roots. Therefore, if the critical values are p and q with p < q, the solution is x < p or x > q.
当求解 ax² + bx + c > 0 且 a > 0 时，抛物线开口向上。表达式大于零的部分在两个根之间的区间之外。因此，若临界值为 p 和 q 且 p < q，解为 x < p 或 x > q。
or instance, x² − x − 6 > 0 gives critical values −2 and 3. The graph of y = x² − x − 6 is ∪ curve is above the x-axis when x < −2 and when x > 3. So the solution is x < −2 or x > 3.
例如，x² − x − 6 > 0 的临界值为 −2 和 3。y = x² − x − 6 的图像为 ∪ 形，与 x 轴相交于 x = −2 和 x = 3。当 x < −2 和 x > 3 时，曲线在 x 轴上方。因此，解为 x < −2 或 x > 3。
7. Determining the Solution Interval for < 0 | 确定小于零的解区间
For ax² + bx + c < 0 with a > 0, the expression is negative only in the interval between the two roots. Hence the solution is p < x < q, where p and q are the critical values and p < q.
对于 ax² + bx + c < 0 且 a > 0，表达式仅在两个根之间的区间为负。因此，解为 p < x < q，其中 p 和 q 为临界值且 p < q。
refore −2 < x < 3. This is often the most tested case, so be careful to write the solution as a single interval, not two separate ones.
仍以 x² − x − 6 < 0 为例，抛物线在 x = −2 和 x = 3 之间位于 x 轴下方。因此解为 −2 < x < 3。这是考试中最常见的题型，因此请务必小心，要将解写成单个区间，而非两个分离的区间。
8. Including Equality – ≥ and ≤ | 包含等号——≥ 和 ≤
elves form part of the solution. The interval notation and number line must reflect this by using closed circles instead of open ones.
当不等式包含等号（≤ 或 ≥）时，临界值本身也是解的一部分。区间记号和数轴表示必须用实心圆而非空心圆来体现这一点。
nd the intervals remains identical; the only difference is the inclusion of the endpoints.
对于 x² − x − 6 ≥ 0，解变为 x ≤ −2 或 x ≥ 3。对于 x² − x − 6 ≤ 0，解则为 −2 ≤ x ≤ 3。区间选择的逻辑完全相同，唯一区别在于端点是否包含在解内。
9. Solving by Factorising – Worked Example | 因式分解法例题
, the parabola opens upwards. The expression is positive outside the roots, so the solution is x < −½ or x > 2.
我们来解 2x² − 3x − 2 > 0。首先因式分解：(2x + 1)(x − 2) > 0。将每个括号设为零求临界值：x = −½ 和 x = 2。由于 a = 2 > 0，抛物线开口向上。表达式在根的外侧为正，因此解为 x < −½ 或 x > 2。
1)(0 − 2) = (1)(−2) = −2 < 0, so the inside interval does not satisfy the inequality > 0.
务必要选取一个测试点，如 x = 0，以确认两根之间的区域确实为负：(2·0 + 1)(0 − 2) = (1)(−2) = −2 < 0，故内侧区间不满足不等式 > 0。
10. Solving by Quadratic Formula – Worked Example | 公式法例题
ise simply, so use the formula with a = 1, b = 4, c = 1. Discriminant: 4² − 4·1·1 = 12, so √12 = 2√3. The roots are x = [−4 ± 2√3] / 2 = −2 ± √3. The critical values are −2 − √3 and −2 + √3. Since a > 0, the parabola is ∪-shaped, so the expression ≤ 0 between the roots: −2 − √3 ≤ x ≤ −2 + √3.
解不等式 x² + 4x + 1 ≤ 0。该二次式无法简单因式分解，因此用求根公式，a = 1, b = 4, c = 1。判别式：4² − 4·1·1 = 12，所以 √12 = 2√3。根为 x = [−4 ± 2√3] / 2 = −2 ± √3。临界值为 −2 − √3 和 −2 + √3。因为 a > 0，抛物线开口向上，所以表达式 ≤ 0 的区间在两根之间：−2 − √3 ≤ x ≤ −2 + √3。
When asked to give answers in exact surd form, leave the roots as −2 ± √3 and use them directly in the inequality. Approximating to decimals is acceptable only when the question specifically requests it.
若要求以精确的根式形式给出答案，将根写作 −2 ± √3 并直接用于不等式中。仅在题目明确要求时才将答案近似为小数。
11. Graphical Representation on Number Lines | 在数轴上的图形表示
E questions often ask you to represent the solution on a number line. Use an open circle (○) for strict inequalities < or >, and a closed circle (●) for ≤ or ≥. Shade the region(s) of x that satisfy the inequality.
IGCSE 题目常要求你在数轴上表示解集。对于严格不等式 < 或 >，使用空心圆 (○)；对于 ≤ 或 ≥，使用实心圆 (●)。将满足不等式的 x 区域涂上阴影。
For x < −½ or x > 2, draw an open circle at −½ and 2, then shade the line extending to the left from −½ and to the right from 2. For −2 − √3 ≤ x ≤ −2 + √3, mark closed circles at both critical values and shade the segment between them.
对于 x < −½ 或 x > 2，在 −½ 和 2 处画空心圆，然后将 −½ 左侧和 2 右侧的线段涂上阴影。对于 −2 − √3 ≤ x ≤ −2 + √3，在两个临界值处标实心圆，并在它们之间的线段上涂阴影。
Always label the critical values on the number line, especially when they involve surds or fractions, to show clearly where the boundaries of the solution are.
一定要在数轴上标注临界值，特别是当它们包含根式或分数时，以清晰显示解的边界位置。
12. Common Mistakes to Avoid | 常见错误须知
One common mistake is forgetting to rearrange the inequality to have zero on one side before solving. Students sometimes attempt to factorise and solve while the constant term remains on the wrong side, leading to incorrect critical values.
一个常见的错误是在求解前忘记将不等式整理为一边为零。学生有时会在常数项仍留在错误的一边时就尝试因式分解和求解，从而导致临界值错误。
Another frequent error is misinterpreting the direction of the inequality after multiplying or dividing by a negative number. Always reverse the inequality sign when multiplying or dividing by a negative quantity. Also, avoid writing the solution to x² < 9 simply as x < 3 or x < −3; the correct solution is −3 < x < 3, which is fundamentally different.
另一个常见错误是在两边乘或除以负数后错误解读不等号方向。一定记住，乘或除以负数时要反转不等号。此外，不要将 x² < 9 的解简单地写为 x < 3 或 x < −3；正确的解是 −3 < x < 3，这有着本质区别。
Finally, when using the quadratic formula, students sometimes mis-copy the signs of b and c, especially when substituting negative values. Write down the formula with substituted values in a separate step to reduce arithmetic slips.
最后，在使用求根公式时，学生有时会在代入时抄错 b 和 c 的符号，尤其在代入负值时更易出错。建议在单独的步骤中写出代入数后的公式，以减少计算失误。
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