📚 Mastering Quadratic Equations | 掌握二次方程
Quadratic equations appear throughout the Edexcel IGCSE Mathematics syllabus, from factorisation and the quadratic formula to graph sketching and real‑world modelling. This article explains every core method in straightforward language, with plenty of examples so you can tackle exam questions confidently.
二次方程遍及 Edexcel IGCSE 数学考纲，从因式分解、求根公式到图像绘制与实际建模均有涉及。本文用清晰的语言讲解每种核心方法，并配有大量例子，帮助读者自信应对考试。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0 where a, b and c are constants and a ≠ 0. The highest power of x is 2, which gives a parabolic graph and often two solutions (roots).
二次方程是能写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。x 的最高次数为 2，图像为抛物线，通常有两个解（根）。
2. Standard Form and the Meaning of Coefficients | 标准形式与系数的意义
Regroup terms so the equation reads ax² + bx + c = 0. The coefficient a controls the width and direction of the parabola, b affects the position of the axis of symmetry, and c gives the y‑intercept. In IGCSE questions, you must often rearrange before solving.
将各项重新排列，使方程呈现 ax² + bx + c = 0。系数 a 决定抛物线的宽窄与开口方向，b 影响对称轴的位置，c 则是 y 轴截距。在 IGCSE 题目中，解题前常常需要先整理成标准形式。
3. Solving Quadratics by Factorisation | 通过因式分解求解
If a quadratic factorises into two linear brackets, set each bracket equal to zero. For example, x² + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0, giving x = −2 or x = −3. Always check by expanding. Edexcel papers frequently test this with non‑monic quadratics (a ≠ 1) as well.
若二次式可分解为两个一次括号，令每个括号等于零即可。例如 x² + 5x + 6 = 0 化为 (x + 2)(x + 3) = 0，得 x = −2 或 x = −3。务必通过展开验算。Edexcel 试卷经常考查非首一型（a ≠ 1）的因式分解。
- Always write the equals‑to‑zero form first | 始终先写成等于零的形式
- Look for a common factor before starting | 开始前先寻找公因子
- Use the 'ac method' when a ≠ 1 | a ≠ 1 时使用“ac 方法”
4. Solving by Completing the Square | 用配方法求解
Rewrite ax² + bx + c in the form a(x + p)² + q. For x² + 6x + 5 = 0, half the coefficient of x gives (x + 3)² − 9 + 5 = 0, so (x + 3)² = 4 and x = −1 or x = −5. This method is essential for deriving the quadratic formula and for finding turning points on graphs.
将 ax² + bx + c 改写为 a(x + p)² + q 的形式。以 x² + 6x + 5 = 0 为例，取 x 系数的一半得 (x + 3)² − 9 + 5 = 0，即 (x + 3)² = 4，故 x = −1 或 x = −5。配方法是推导求根公式和寻找图像顶点的基础。
5. The Quadratic Formula | 求根公式
When factorisation is awkward, use x = (−b ± √b² − 4ac) / 2a. This formula always works. Plug in the values of a, b and c and simplify step by step. On IGCSE papers, you must memorise the formula — it is not given on the formula sheet for the Higher tier unless stated otherwise, so learn it by heart.
当因式分解不便时，使用 x = (−b ± √b² − 4ac) / 2a。该公式始终有效。代入 a、b、c 的值并逐步化简。在 IGCSE 考试中需熟记该公式——除非特别说明，公式表上未必提供，务必牢记。
6. The Discriminant (Δ = b² − 4ac) | 判别式分析
The value of b² − 4ac when Δ < 0. Edexcel often asks you to find the number of solutions without solving fully.
b² − 4ac 的值可以判断根的情况：Δ > 0 有不等实根，Δ = 0 有等根，Δ < 0 无实根。Edexcel 常要求在不彻底解方程的情况下判断解的个数。
| Condition | 条件 | Number of real roots | 实根个数 | Graph behaviour | 图像表现 |
|---|---|---|
| b² − 4ac > 0 | 2 distinct | 两个不等 | Crosses x‑axis twice | 与 x 轴交于两点 |
| b² − 4ac = 0 | 1 repeated | 一个等根 | Touches x‑axis | 与 x 轴相切 |
| b² − 4ac < 0 | 0 real | 无实根 | Does not meet x‑axis | 不与 x 轴相交 |
7. Sketching Quadratic Graphs | 绘制二次函数图像
Identify the y‑ ), and the turning point (completed square or axis of symmetry x = −b/2a). Draw a smooth U‑shaped curve if a > 0, or an ∩‑shaped curve if a < 0. Always label intercepts and the vertex on your sketch.
确定 y 轴截距（c）、根（通过因式分解或公式）以及顶点（配方得到或对称轴 x = −b/2a）。若 a > 0 绘制平滑的 U 形曲线，若 a < 0 绘制 ∩ 形曲线。草图上须标出截距与顶点坐标。
8. Solving Quadratic Inequalities | 解二次不等式
For an inequality such as x² − 4x + 3 ≤ 0 (x = 1 and x = 3). Sketch the parabola and shade the region where the curve is below the x‑axis. The solution is 1 ≤ x ≤ 3. Always use a graph to check the direction of the inequality.
对于 x² − 4x + 3 ≤ 0 这类不等式，先解对应方程求出临界值（x = 1 与 x = 3）。画出抛物线草图，并将曲线在 x 轴下方的部分涂上阴影，得到解集 1 ≤ x ≤ 3。务必用图像检验不等号方向。
9. Worded Problems and Real‑life Applications | 文字题与实际应用
Translate the scenario into a quadratic equation. Typical IGCSE contexts include area of a rectangle, projectile height, or profit optimisation. Define your unknown, form the equation, solve it, and discard solutions that do not fit the context (e.g. negative lengths).
将实际问题转化为二次方程。典型的 IGCSE 情境包括矩形面积、抛射高度或利润最大化。设出未知数、建立方程、求解并舍去不符合实际的解（如长度为负）。
- Let the unknown be x and express other quantities in terms of x | 设未知数为 x，用 x 表示其他量
- Form the equation from the relationship given | 根据给定关系列出方程
- Interpret both solutions and check feasibility | 解释两个解并检查合理性
10. Common Pitfalls and How to Avoid Them | 常见错误与避错方法
Forgotten negatives when factorising, arithmetic slips inside the square root, and forgetting to set the equation to zero before solving are among the most frequent mistakes. Practise writing each step clearly. When using the quadratic formula, double‑check that you substituted b correctly, especially when b is negative.
因式分解时漏掉负号、根号内部计算错误以及解方程前忘记移项成零是最高频的失误。应练习每一步写清楚。使用求根公式时，尤其当 b 为负值时，务必反复核对其代入是否正确。
11. Quick Reference: Comparing Methods | 方法速查对比
| Method | 方法 | When to use | 何时使用 | Example | 示例 |
|---|---|---|
| Factorisation | Equation factorises easily | 方程易于因式分解 | x² − 7x + 12 = 0 → (x−3)(x−4)=0 |
| Completing square | Need vertex or exact form | 需要顶点或精确形式 | x² + 10x + 9 → (x+5)²−16 |
| Quadratic formula | Any quadratic, especially messy coefficients | 任意二次式，系数复杂时 | 2x²+3x−4=0 → x = [−3±√9+32]/4 |
12. Practice and Exam Tips | 练习与应试建议
Edexcel IGCSE questions often mix algebraic manipulation with graph work. Practise past papers under timed conditions. When a question asks for answers 'in surd form' or 'to 3 significant figures', pay close attention — the quadratic formula will usually be needed. Show your working clearly; method marks can make a big difference.
Edexcel IGCSE 题目常将代数操作与图像综合考查。应在限时条件下练习往年真题。当题目要求“以根式形式”或“保留三位有效数字”时，务必注意——通常需要使用求根公式。书写步骤清晰，方法分往往能带来很大帮助。
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