📚 Mastering Quadratics: Equations, Inequalities and Graphs | 精通二次函数：方程、不等式与图像
Quadratics are one of the most heavily tested topics in IGCSE Mathematics. From solving equations to sketching graphs and tackling inequalities, a solid understanding of quadratic functions unlocks high-scoring marks across both Paper 2 and Paper 4.
二次函数是IGCSE数学中考查最频繁的主题之一。从解方程到画图，再到处理不等式，扎实掌握二次函数是你在Paper 2和Paper 4中取得高分的关键。
1. The Standard Form of a Quadratic | 二次函数的标准形式
A quadratic expression is written in the general form ax² + bx + c, where a, b and c are constants and a ≠ 0. The graph of a quadratic function is a smooth U-shaped curve called a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
二次表达式的一般形式为ax² + bx + c，其中a、b、c为常数，且a ≠ 0。二次函数的图像是一条光滑的U形曲线，称为抛物线。当a > 0时，抛物线开口向上；当a < 0时，开口向下。
f(x) = ax² + bx + c, a ≠ 0
Key features of the graph include the y-intercept (0, c), the axis of symmetry x = −b/(2a), and the vertex (turning point) of the parabola.
图像的关键特征包括y轴截距(0, c)、对称轴x = −b/(2a)，以及抛物线的顶点（转向点）。
2. Solving by Factorisation | 因式分解法
Factorisation is often the quickest method when a quadratic has simple integer roots. To solve x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5. These are 2 and 3, so the equation becomes (x + 2)(x + 3) = 0.
当二次方程具有简单的整数根时，因式分解通常是最快的方法。要解x² + 5x + 6 = 0，找到两个相乘为6、相加为5的数，即2和3。于是方程变为(x + 2)(x + 3) = 0。
(x + 2)(x + 3) = 0 → x = −2 or x = −3
- If the coefficient of x² is not 1, use the method of factoring by grouping or the “product-sum” technique.
- 如果x²的系数不为1，可以使用分组分解法或“积-和”技巧。
- Always expand your brackets back to check your factors are correct.
- 始终将括号展开回原式，以检查因式是否正确。
3. The Quadratic Formula | 公式法
When a quadratic cannot be factorised easily, the quadratic formula gives a reliable solution for any equation of the form ax² + bx + c = 0.
当二次方程不易因式分解时，公式法为任意ax² + bx + c = 0形式的方程提供了可靠的解法。
x = (−b ± √(b² − 4ac)) / (2a)
Substitute the values of a, b and c carefully. Use a calculator to evaluate the square root, then simplify both solutions separately. For example, solve 2x² − 4x − 3 = 0:
代入a、b、c的值时要仔细。使用计算器计算平方根，然后分别化简两个解。例如，解2x² − 4x − 3 = 0：
x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4
This gives x ≈ 2.58 or x ≈ −0.58. Always round to an appropriate degree of accuracy.
得到x ≈ 2.58或x ≈ −0.58。记得将答案保留到合适的精度。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This form reveals the vertex directly: the turning point is at (−p, q).
配方法将二次式改写为a(x + p)² + q的形式。这种形式直接揭示了顶点：转向点为(−p, q)。
x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4
The process: halve the coefficient of x, square it, then adjust the constant. For 2x² − 8x + 7, first factor out the 2 from the x terms:
步骤：将x系数的半平方，然后调整常数。对于2x² − 8x + 7，先从含x的项中提取因数2：
2[(x − 2)² − 4] + 7 = 2(x − 2)² − 1
Hence the vertex is (2, −1). This form also makes solving equations and finding minimum or maximum values straightforward.
因此顶点为(2, −1)。这种形式也简化了方程的求解以及最大值、最小值的确定。
5. The Discriminant | 判别式
The discriminant, Δ = b² − 4ac, determines the nature of the roots of a quadratic equation without solving it completely.
判别式Δ = b² − 4ac决定了二次方程根的性质，无需完整求解方程。
| Discriminant | Nature of Roots |
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | No real roots |
判别式与根的关系如下：Δ > 0时有两个不同的实数根；Δ = 0时有一个重根；Δ < 0时没有实数根。这个结论在选择题和简答题中经常出现。
6. Solving Word Problems | 应用题
Many IGCSE problems require setting up a quadratic equation from a real-world context. Common scenarios include area problems, number puzzles and projectile motion.
许多IGCSE题目需要根据实际情境建立二次方程。常见的情境包括面积问题、数字谜题和抛体运动。
Example: A rectangle has length 3 cm longer than its width. Its area is 54 cm². Find the width.
例题：一个长方形的长比宽长3 cm，面积为54 cm²。求宽。
x(x + 3) = 54 → x² + 3x − 54 = 0
Factorising gives (x + 9)(x − 6) = 0, so x = 6 (reject x = −9 as a length cannot be negative). Always check whether negative solutions are valid in context.
因式分解得到(x + 9)(x − 6) = 0，因此x = 6（舍去x = −9，因为长度不能为负）。务必检查负数解在具体情境中是否合理。
7. Quadratic Inequalities | 二次不等式
To solve x² − 5x + 6 < 0, first factorise: (x − 2)(x − 3) < 0. The critical values are x = 2 and x = 3. Sketch the parabola or test intervals to determine where the curve is below the x-axis.
解x² − 5x + 6 < 0时，先因式分解为(x − 2)(x − 3) < 0。临界值为x = 2和x = 3。通过画抛物线草图或测试区间，判断曲线位于x轴下方的部分。
2 < x < 3
For a > 0, the inequality ax² + bx + c < 0 is satisfied between the roots, while ax² + bx + c > 0 is satisfied outside the roots. Remember to flip the inequality sign is not needed here — that rule applies only to multiplication by negatives.
当a > 0时，ax² + bx + c < 0的解在两根之间；而ax² + bx + c > 0的解在两根之外。注意此处不需要翻转不等号——该规则仅适用于乘以负数时。
8. Graphs and the Intersection of Lines | 图像与直线交点
Solving a quadratic equation graphically means finding the x-coordinates where the parabola crosses the x-axis. The equation ax² + bx + c = 0 corresponds to y = 0 on the graph.
图解法求解二次方程，就是找出抛物线与x轴交点的x坐标。方程ax² + bx + c = 0对应于图像上y = 0的位置。
When a straight line intersects a parabola, their intersection points are solutions to the simultaneous equations. Substitute y = mx + k into the quadratic to obtain a new quadratic in x. The discriminant then tells you the number of intersection points:
当直线与抛物线相交时，其交点就是联立方程的解。将y = mx + k代入二次函数，得到关于x的新二次方程。判别式告诉我们交点的数量：
- Δ > 0: two intersection points | 两个交点
- Δ = 0: the line is tangent to the curve | 直线与曲线相切
- Δ < 0: no intersection | 没有交点
9. The Vertex Form and Transformations | 顶点式与图像变换
The vertex form y = a(x − h)² + k allows you to sketch a parabola quickly. The graph of y = x² is shifted horizontally by h units and vertically by k units. The sign of a determines orientation; the magnitude of a affects the steepness.
顶点式y = a(x − h)² + k可以帮助你快速画抛物线。y = x²的图像水平平移h个单位、垂直平移k个单位。a的符号决定开口方向，a的大小影响抛物线的陡峭程度。
y = (x − 3)² + 2 → vertex at (3, 2)
In IGCSE exams, you may be asked to describe a transformation: “Translate y = x² by 3 units in the positive x-direction and 2 units in the positive y-direction.” Being precise with direction and magnitude earns full marks.
在IGCSE考试中，可能会要求你描述图像的变换：“将y = x²沿x轴正方向平移3个单位，沿y轴正方向平移2个单位。”准确描述方向和距离，才能获得满分。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Many students lose marks on quadratics due to small but avoidable errors. Here are the most common pitfalls and how to avoid them.
很多学生在二次函数题目上失分，往往是由一些细小但可以避免的错误造成的。以下是最常见的陷阱以及如何避免它们。
- Sign errors: when using the quadratic formula, remember (−b) is positive if b is negative.
- 符号错误：使用公式法时，若b为负数，则(−b)为正数。
- Forgetting a = 0 is not allowed: if a = 0, the expression is linear, not quadratic.
- 忘记a ≠ 0：当a = 0时，表达式是一次式，而非二次式。
- Leaving answers as surds when the question asks for decimals — and vice versa.
- 题目要求小数却保留根号，或题目要求精确值却写成小数。
- Not rejecting extraneous or negative solutions in word problems.
- 在应用题中未舍去不合理的负数解或增根。
11. Practice Checklist | 练习清单
Before your IGCSE exam, make sure you can confidently perform each of the following skills. Tick them off as you revise.
在IGCSE考试前，请确保你能自信地完成以下每一项技能。复习时逐项勾选。
| Skill | Confident? |
| Factorise ax² + bx + c (including a ≠ 1) | ☐ |
| Solve using the quadratic formula | ☐ |
| Complete the square and state the vertex | ☐ |
| Use the discriminant to determine root types | ☐ |
| Solve quadratic inequalities | ☐ |
| Sketch graphs from vertex form | ☐ |
Practise mixed questions from past papers to build speed. Timing is critical: a quadratics question worth 4 marks should take no more than 4–5 minutes.
通过刷历年真题中的混合题型来提升速度。时间管理至关重要：一道4分的二次函数题不应超过4至5分钟。
12. Summary | 总结
Quadratic equations, inequalities and graphs form a cohesive topic that connects algebra, geometry and problem-solving. Master the three solving methods — factorisation, formula and completing the square — and understand what the discriminant tells you. Sketch graphs accurately, label key features, and always interpret your answers in context.
二次方程、不等式和图像将代数、几何与问题解决能力紧密联系在一起。掌握三种解法——因式分解、公式法和配方法，并理解判别式的含义。准确画图、标注关键特征，并始终结合情境解读答案。
Practice daily, check your working, and quadratics will become one of your strongest topics.
每日练习、检查过程，二次函数将成为你最擅长的考点之一。
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