📚 Exercise 3D: Quadratic Inequalities | Exercise 3D：二次不等式
In the AQA A-Level Mathematics curriculum, Chapter 3 focuses on equations and inequalities. Exercise 3D builds on the foundation of solving quadratic equations to introduce one of the most frequently tested skills in Pure Mathematics: solving quadratic inequalities. This exercise bridges algebraic manipulation and graphical reasoning, and mastering it is essential for success in later topics such as optimisation, integration and numerical analysis.
在 AQA 进阶数学课程中，第三章聚焦于方程与不等式。Exercise 3D 在解二次方程的基础上，引入了纯数学中考查频率最高的技能之一：解二次不等式。本练习将代数运算与图像推理相结合，掌握它对于后续的优化问题、积分和数值分析等专题至关重要。
1. What Is a Quadratic Inequality? | 什么是二次不等式？
A quadratic inequality is an inequality that involves a quadratic expression. In standard form, it can be written as ax² + bx + c > 0, ax² + bx + c ≥ 0, ax² + bx + c < 0 or ax² + bx + c ≤ 0, where a, b and c are constants and a ≠ 0. The solution set of such an inequality is usually an interval, or the union of two intervals, on the real number line — not a single value or a pair of values as with a quadratic equation.
二次不等式是涉及二次表达式的不等式。其标准形式可写作 ax² + bx + c > 0、ax² + bx + c ≥ 0、ax² + bx + c < 0 或 ax² + bx + c ≤ 0，其中 a、b、c 为常数且 a ≠ 0。此类不等式的解集通常是实数轴上的一段区间，或两段区间的并集——而非像二次方程那样是一个值或一对值。
For example, the inequality x² − 5x + 6 > 0 does not ask for the roots of x² − 5x + 6 = 0; it asks for every real value of x that makes the quadratic expression positive. The roots 2 and 3 are merely the boundary points that separate positive regions from negative regions.
例如，不等式 x² − 5x + 6 > 0 要求的并不是方程 x² − 5x + 6 = 0 的根，而是使该二次表达式为正的所有实数 x。根 2 和 3 仅仅是分隔正区域与负区域的边界点。
2. Rearranging into Standard Form | 整理为标准形式
Before solving a quadratic inequality, always collect all non-zero terms on one side of the inequality sign. For instance, to solve x² < 2x + 3, subtract 2x + 3 from both sides to obtain x² − 2x − 3 < 0. Only when the inequality is in standard form can you reliably identify the corresponding quadratic equation and its roots.
在求解二次不等式之前，务必将所有非零项移到不等式的一侧。例如，解 x² < 2x + 3 时，两边同时减去 2x + 3，得到 x² − 2x − 3 < 0。只有当不等式处于标准形式时，才能可靠地确定对应的二次方程及其根。
If the coefficient of x² is negative, multiply both sides of the inequality by −1 and reverse the inequality sign. This step is critical: multiplying or dividing an inequality by a negative number flips the direction of the inequality. After this step, the leading coefficient is positive, which makes the graph a U-shaped parabola and allows you to apply the standard sign rules without confusion.
若 x² 的系数为负数，则将不等式两边同时乘以 −1，并反向不等式号。这一步至关重要：不等式两边乘以或除以负数时，不等式方向必须反转。完成此步后，首项系数为正，图像为开口向上的抛物线，此时可以放心应用标准的符号判定规则。
Example: −3x² + 6x < 0 → 3x² − 6x > 0
Here the direction of the inequality has been reversed because both sides were multiplied by −1. The new quadratic 3x² − 6x has a positive leading coefficient and is much easier to analyse graphically.
此处两边同时乘以 −1，因此不等式方向发生了反转。新的二次表达式 3x² − 6x 具有正的首项系数，从图像角度分析要容易得多。
3. Step 1: Solve the Corresponding Equation | 第一步：解对应方程
Replace the inequality sign with an equals sign and solve the resulting quadratic equation. Factorisation, the quadratic formula or completing the square are all valid methods. The two roots, often labelled x₁ and x₂, are called the critical values of the inequality. These values mark the points where the graph of y = ax² + bx + c crosses or touches the x-axis, and hence where the sign of the quadratic expression can change.
将不等式号换成等号，并解出相应的二次方程。因式分解、二次求根公式或配方法均可使用。方程的两个根通常记为 x₁ 和 x₂，称为不等式的临界值。这些值标记了 y = ax² + bx + c 的图像穿过或与 x 轴相切的位置，也就是二次表达式符号可能发生变化的点。
Consider the inequality x² − 5x + 6 > 0. Begin by solving:
考虑不等式 x² − 5x + 6 > 0。首先求解：
x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x₁ = 2, x₂ = 3
The critical values are 2 and 3. These two points split the real number line into three disjoint intervals: (−∞, 2), (2, 3) and (3, ∞). In each of these intervals, the sign of x² − 5x + 6 remains constant, so testing one value per interval tells you the sign for the whole interval.
临界值为 2 和 3。这两个点将实数轴分成三个互不相交的区间：(−∞, 2)、(2, 3) 和 (3, ∞)。在每个区间内，x² − 5x + 6 的符号保持恒定，因此每段区间内取一个测试值，即可判断整段区间的符号。
4. Step 2: Sketch or Construct a Sign Diagram | 第二步：画图或绘制符号图
With a positive leading coefficient, the graph of y = ax² + bx + c is a U-shaped parabola. If the equation has two distinct roots x₁ and x₂ (with x₁ < x₂), the curve lies below the x-axis between the roots and above the x-axis outside the roots. This simple observation gives you the sign pattern directly: negative in the middle, positive on both ends.
当首项系数为正时，y = ax² + bx + c 的图像是开口向上的抛物线。若方程有两个不同的根 x₁ 和 x₂（且 x₁ < x₂），则曲线在两根之间位于 x 轴下方，在两根之外位于 x 轴上方。这一直观观察直接给出了符号模式：中间为负，两端为正。
A sign diagram achieves the same result without drawing a full graph. Mark the critical values in increasing order on a number line, then substitute a convenient test value from each interval into the quadratic expression to determine its sign.
符号图无需画出完整图像即可达到同样效果。在数轴上按递增顺序标出临界值，然后从每个区间选取一个方便的测试值代入二次表达式，判断其符号。
| Interval | (−∞, 2) | (2, 3) | (3, ∞) |
| Test value | x = 0 | x = 2.5 | x = 4 |
| x² − 5x + 6 | 6 | −0.25 | 2 |
| Sign | + | − | + |
The sign diagram shows clearly that the quadratic expression is positive on (−∞, 2) and (3, ∞), and negative on (2, 3). This table-based method is quick, accurate and easy to reproduce under exam conditions.
符号图清楚地显示，二次表达式在 (−∞, 2) 和 (3, ∞) 上为正，在 (2, 3) 上为负。这种表格化的方法快速、准确，而且在考试条件下易于复现。
5. Step 3: Read Off the Solution Set | 第三步：读出解集
Once you know the sign of the quadratic in each interval, select the intervals that satisfy the original inequality. For x² − 5x + 6 > 0, the quadratic is positive on (−∞, 2) and (3, ∞), so the solution is x < 2 or x > 3. In set notation this is written as {x : x < 2} ∪ {x : x > 3}, or equivalently (−∞, 2) ∪ (3, ∞).
一旦知道每个区间内二次函数的符号，选取满足原不等式的区间即可。对于 x² − 5x + 6 > 0，二次函数在 (−∞, 2) 和 (3, ∞) 上为正，所以解为 x < 2 或 x > 3。用集合记号写作 {x : x < 2} ∪ {x : x > 3}，等价地写作 (−∞, 2) ∪ (3, ∞)。
For x² − 5x + 6 < 0, the quadratic is negative on (2, 3), so the solution is 2 < x < 3, or simply (2, 3). Notice that for a positive leading coefficient, the general rule is always the same: "greater than zero" gives two outside intervals, while "less than zero" gives one middle interval between the roots.
对于 x² − 5x + 6 < 0，二次函数在 (2, 3) 上为负，所以解为 2 < x < 3，简写为 (2, 3)。注意，对于正的首项系数，一般规律始终一致："> 0″ 给出两段外部区间，而 “< 0" 给出两根本身之间的一段中间区间。
Summary: if a > 0 and x₁ < x₂ are the roots, then ax² + bx + c > 0 ⇔ x < x₁ or x > x₂; and ax² + bx + c < 0 ⇔ x₁ < x < x₂.
This summary can be memorised as a quick reference, but always check the leading coefficient sign first. If a < 0, multiply by −1 before applying the rule.
可将这一总结作为快捷参考加以记忆，但务必先确认首项系数的符号。若 a < 0，先乘以 −1 再套用规则。
6. Strict vs Non-Strict Inequalities | 严格与非严格不等式
For strict inequalities > and <, the critical values are excluded from the solution set. On a number line, represent excluded endpoints with an open circle. For non-strict inequalities ≥ and ≤, the critical values are included, so use a closed circle on a number line. In interval notation, round brackets ( ) indicate excluded endpoints and square brackets [ ] indicate included endpoints.
对于严格不等式 > 和 <，临界值不包含在解集之内。在数轴上，排除的端点用空心圆表示。对于非严格不等式 ≥ 和 ≤，临界值包含在解集中，在数轴上用实心圆表示。在区间记号中，圆括号 ( ) 表示端点被排除，方括号 [ ] 表示端点被包含。
Compare the four possibilities for x² − 5x + 6:
比较 x² − 5x + 6 的四种情况：
| Inequality |
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