📚 Solving Quadratic Inequalities by Graphing | 通过图像解二次不等式
ic methods are useful, graphing provides a clear and intuitive way to determine the solution set. In this article, we will explore how to solve quadratic inequalities by sketching the corresponding parabola and interpreting the regions where the graph lies above or below the x-axis.
二次不等式是 IGCSE 数学中的一个重要考点。二次不等式涉及一个二次表达式和不等号：<、>、≤ 或 ≥。虽然代数方法很实用，但图像法提供了一种清晰直观的求解方式。本文将探讨如何通过画出对应的抛物线，并判断图像在 x 轴上方或下方的区间，来解二次不等式。
1. Understanding Quadratic Inequalities | 理解二次不等式
A quadratic inequality can be written in the general form:
ax² + bx + c < 0, ax² + bx + c > 0, ax² + bx + c ≤ 0, or ax² + bx + c ≥ 0
where a, b, and c are real numbers and a ≠ 0. The solution set consists of all x-values that satisfy the inequality. Unlike a quadratic equation, which usually has at most two solutions, a quadratic inequality typically has a range of solutions, often expressed as intervals.
二次不等式的一般形式可写成：
ax² + bx + c < 0、ax² + bx + c > 0、ax² + bx + c ≤ 0 或 ax² + bx + c ≥ 0
其中 a、b、c 是实数，且 a ≠ 0。解集是满足不等式的所有 x 值。与通常最多只有两个解的二次方程不同，二次不等式通常有一系列解，多以区间表示。
2. Key Concepts: Roots and the x-axis | 关键概念：根与 x 轴
the x-axis into intervals. Within each interval, the quadratic expression is either always positive or always negative, because the parabola can only change sign when it crosses the x-axis.
用图像法解二次不等式时，先观察 y = ax² + bx + c 的图像。抛物线与 x 轴交点的横坐标就是根，通过令 ax² + bx + c = 0 求出。这些根把 x 轴分成若干区间。在每个区间内，二次表达式的符号恒定为正或恒定为负，因为抛物线只有在穿过 x 轴时才会改变符号。
Consider the quadratic equation:
x² − 5x + 6 = 0
hese roots split the x-axis into three intervals: x < 2, 2 < x < 3, and x > 3.
例如二次方程：
x² − 5x + 6 = 0
因式分解得 (x − 2)(x − 3) = 0，所以根是 x = 2 和 x = 3。这两个根把 x 轴分成三个区间：x < 2、2 < x < 3 和 x > 3。
3. Parabola Shape and Leading Coefficient | 抛物线开口方向与二次项系数
The sign of the leading coefficient a determines the direction of the parabola's opening. If a > 0, the parabola opens upward like a U-shape, and the quadratic expression is positive outside the roots and negative between the roots. If a < 0, the parabola opens downward like an inverted U, and the expression is negative outside the roots and positive between the roots.
二次项系数 a 的符号决定抛物线的开口方向。若 a > 0，抛物线开口向上，像 U 形，二次表达式在两根之外为正、两根之间为负。若 a < 0，抛物线开口向下，像倒 U 形，表达式在两根之外为负、两根之间为正。
This relationship is central to solving inequalities graphically. By drawing the parabola, we can immediately see which parts of the curve are above the x-axis (y > 0) and which are below (y < 0).
这个关系是图像法解不等式的核心。画出抛物线后，我们可以直接看出曲线哪些部分在 x 轴上方（y > 0），哪些部分在 x 轴下方（y < 0）。
4. Solving ax² + bx + c > 0 | 解 ax² + bx + c > 0
For an inequality of the form ax² + bx + c > 0, we look for x-values where the quadratic expression is positive, i.e., where the graph is strictly above the x-axis. If a > 0, the graph is above the x-axis to the left of the smaller root and to the right of the larger root.
对于形如 ax² + bx + c > 0 的不等式，我们需要找到二次表达式为正的 x 值，即图像严格位于 x 轴上方的部分。若 a > 0，图像在较小根的左侧和较大根的右侧位于 x 轴上方。
3. The graph is above the x-axis when x < 2 or x > 3. Therefore the solution set is:
x < 2 or x > 3
我们用图像法解 x² − 5x + 6 > 0。抛物线开口向上，根为 x = 2 和 x = 3。当 x < 2 或 x > 3 时，图像在 x 轴上方。因此解集为：
x < 2 或 x > 3
In interval notation, this is written as (−∞, 2) ∪ (3, ∞). The union symbol ∪ combines separate intervals.
用区间表示，即 (−∞, 2) ∪ (3, ∞)。并集符号 ∪ 表示两个不相连的区间。
5. Solving ax² + bx + c < 0 | 解 ax² + bx + c < 0
For ax² + bx + c < 0, we look for x-values where the quadratic expression is negative, i.e., where the graph is strictly below the x-axis. For an upward-opening parabola with real roots, this is the region between the two roots.
对于 ax² + bx + c < 0，我们寻找二次表达式为负的 x 值，即图像严格位于 x 轴下方的部分。对开口向上且有实根的抛物线，这个区域位于两个根之间。
Using the same example, x² − 5x + 6 < 0 has solution 2 < x < 3. The graph lies below the x-axis only between the roots.
用同样的例子，x² − 5x + 6 < 0 的解是 2 < x < 3。图像只在两根之间位于 x 轴下方。
In interval notation: (2, 3).
区间表示为 (2, 3)。
6. Solving Inequalities with ≤ or ≥ | 解含 ≤ 或 ≥ 的不等式
he roots are included in the solution set because at the roots the quadratic expression equals zero. Therefore we use closed intervals or include the endpoint values.
当不等式中包含 ≤ 或 ≥ 时，根包含在解集中，因为在根处二次表达式的值等于零。因此我们使用闭区间或包含端点值。
olution is 2 ≤ x ≤ 3. Note that both 2 and 3 satisfy the inequality because 0 ≤ 0 is true.
对于 x² − 5x + 6 ≤ 0，解为 2 ≤ x ≤ 3。注意 2 和 3 都满足不等式，因为 0 ≤ 0 成立。
Similarly, x² − 5x + 6 ≥ 0 has solution x ≤ 2 or x ≥ 3. In interval notation, (−∞, 2] ∪ [3, ∞).
同样，x² − 5x + 6 ≥ 0 的解为 x ≤ 2 或 x ≥ 3。区间表示为 (−∞, 2] ∪ [3, ∞)。
7. Example: Negative Leading Coefficient | 示例：二次项系数为负
Consider the inequality −x² + 4x − 3 > 0. First multiply both sides by −1 and reverse the inequality sign to make the leading coefficient positive: x² − 4x + 3 < 0. Factorising gives (x − 1)(x − 3) < 0. The parabola opens upward, so the solution is 1 < x < 3.
考虑不等式 −x² + 4x − 3 > 0。先将两边乘以 −1 并反转不等号，使二次项系数为正：x² − 4x + 3 < 0。因式分解得 (x − 1)(x − 3) < 0。抛物线开口向上，所以解为 1 < x < 3。
d, and it is above the x-axis between its roots x = 1 and x = 3, giving the same solution.
或者直接画出原抛物线 y = −x² + 4x − 3。它开口向下，在其根 x = 1 和 x = 3 之间的部分位于 x 轴上方，得到同样的解。
This example illustrates that multiplying by a negative number reverses the inequality sign, a crucial step in algebraic manipulation.
这个例子说明，乘以负数会反转不等号，这是代数运算中的关键步骤。
8. Graphical Sketching Method | 图像画法步骤
To solve any quadratic inequality by graphing, follow these steps:
用图像法解任意二次不等式，可按以下步骤进行：
Step 1: Rewrite the inequality in standard form ax² + bx + c < 0, > 0, ≤ 0, or ≥ 0.
第一步：将不等式改写为标准形式 ax² + bx + c < 0、> 0、≤ 0 或 ≥ 0。
Step 2: ion ax² + bx + c = 0 using factorisation, completing the square, or the quadratic formula.
第二步：通过因式分解、配方法或求根公式求出方程 ax² + bx + c = 0 的根。
Step 3: Determine the direction of the parabola based on the sign of a. Sketch the graph, marking the roots on the x-axis.
第三步：根据 a 的符号确定抛物线开口方向，画出图像并在 x 轴上标出根。
Step 4: Identify the regions where the graph is above or below the x-axis, and write the solution set accordingly.
第四步：确定图像在 x 轴上方或下方的区间，写出相应的解集。
A well-drawn sketch does not need to be perfectly to scale, but it must show the correct shape, the roots, and the axis of symmetry.
草图不需要完全按比例绘制，但必须显示正确的开口方向、根的位置以及对称轴。
9. Distinct Roots, Equal Roots, and No Real Roots | 两个不等实根、两个相等实根与无实根
repeated root, and the parabola touches the x-axis at a single point without crossing it.
判别式 Δ = b² − 4ac 决定实根的个数。若 Δ > 0，有两个不等实根，x 轴被分成三个区间。若 Δ = 0，有一个重根，抛物线与 x 轴相切于一点但不穿过。
the square is never negative. The inequality x² − 4x + 4 ≥ 0 is true for all real numbers.
例如，x² − 4x + 4 = 0 有重根 x = 2。不等式 x² − 4x + 4 > 0 对所有 x ≠ 2 都成立，因为平方不可能为负。而不等式 x² − 4x + 4 ≥ 0 对所有实数都成立。
If Δ < 0, the
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