📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程精讲
Welcome to this IGCSE Mathematics revision guide on quadratic equations. Quadratic equations form a central part of the algebra syllabus, and they appear frequently in both Core and Extended papers. In this article, we will review the standard form, factorisation, completing the square, the quadratic formula, the discriminant, graphing, and real-world applications. Each section pairs a concise English explanation with a Chinese translation to support bilingual learners.
欢迎阅读本篇 IGCSE 数学二次方程复习指南。二次方程是代数教学大纲的核心内容，在 Core 和 Extended 试卷中都频繁出现。本文将复习标准式、因式分解法、配方法、求根公式、判别式、图像以及实际应用。每个小节都提供简洁的英文讲解和对应的中文翻译，帮助双语学习者理解。
1. What is a Quadratic Equation? | 什么是二次方程？
here a, b and c are constants and a ≠ 0. If a were zero, the equation would become linear rather than quadratic. Quadratic equations can have two real solutions, one repeated real solution, or no real solutions depending on the discriminant.
二次方程是次数为二的多项式方程。它可以写成一般形式 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。如果 a 等于零，方程就变成一次方程而不是二次方程。根据判别式的不同，二次方程可能有两个实根、一个重复实根或没有实根。
ax² + bx + c = 0, where a ≠ 0
2. Standard Form and Key Terms | 标准式与关键术语
ading coefficient, b is the linear coefficient, and c is the constant term. The terms root and solution both refer to values of x that satisfy the equation. The expression ax² + bx + c is also called a quadratic expression.
在求解之前，应先将方程写成标准式：ax² + bx + c = 0。系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。根和解都指满足方程的 x 值。表达式 ax² + bx + c 也称为二次式。
- a is the quadratic coefficient | a 是二次项系数
- b is the linear coefficient | b 是一次项系数
- c is the constant term | c 是常数项
3. Solving by Factorisation | 因式分解法求解
x = -2 or x = -3. Always check that the original equation equals zero before factorising.
当二次式可以写成两个一次因式的乘积时，因式分解法通常是最快的方法。例如，x² + 5x + 6 = 0 可以分解为 (x + 2)(x + 3) = 0。令每个因式等于零，得到 x = -2 或 x = -3。分解前务必确认原方程等于零。
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = -2 or x = -3
4. Solving by Completing the Square | 配方法求解
= -2. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 7. This gives (x + 3)² = 7, so x = -3 ± √7. This method is especially useful for deriving the quadratic formula and for finding the vertex of a parabola.
配方法将二次方程转化为 (x + p)² = q 的形式。从 x² + 6x + 2 = 0 开始，将常数移到右边：x² + 6x = -2。两边加上 (6/2)² = 9：x² + 6x + 9 = 7。得到 (x + 3)² = 7，因此 x = -3 ± √7。配方法在推导求根公式和求抛物线顶点时尤其有用。
x² + 6x + 2 = 0 → (x + 3)² = 7 → x = -3 ± √7
5. The Quadratic Formula | 求根公式
adratic formula gives the solutions of ax² + bx + c = 0 directly. The formula is x = (-b ± √b² - 4ac) / 2a. It works for every quadratic equation, even when factorisation is difficult or impossible. When using the formula, substitute the values of a, b and c carefully, and simplify the square root if possible.
求根公式直接给出 ax² + bx + c = 0 的解。公式为 x = (-b ± √b² - 4ac) / 2a。该方法适用于所有二次方程，即使因式分解困难或无法进行。使用公式时，要仔细代入 a、b、c 的值，并尽可能化简平方根。
x = (-b ± √b² - 4ac) / 2a
6. Discriminant and Nature of Roots | 判别式与根的性质
there is one repeated real root. If b² - 4ac < 0, there are no real roots, but two complex roots.
判别式是求根公式中平方根下的表达式 b² - 4ac。它让我们无需解方程就能判断根的性质。如果 b² - 4ac > 0，方程有两个不同实根；如果 b² - 4ac = 0，有一个重复实根；如果 b² - 4ac < 0，没有实根，但有两个复数根。
| Discriminant | 判别式 | Nature of roots | 根的性质 |
|---|---|
| b² - 4ac > 0 | Two distinct real roots | 两个不同实根 |
| b² - 4ac = 0 | One repeated real root | 一个重复实根 |
| b² - 4ac < 0 | No real roots | 无实根 |
7. Graphical Interpretation | 图像解析
an be found by completing the square or by using x = -b / 2a. The y-intercept is always c.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上；如果 a < 0，开口向下。方程 ax² + bx + c = 0 的解对应图像与 x 轴的交点。顶点可通过配方法或用 x = -b / 2a 求出。y 轴截距始终为 c。
Vertex x-coordinate: x = -b / 2a
8. Applications and Word Problems | 应用题与文字题
Quadratic equations often model real-world situations such as projectile motion, area, and profit. In a typical problem, you must define the unknown, form a quadratic equation, solve it, and interpret the answer in context. Discard negative or unrealistic solutions when the variable represents a physical quantity.
二次方程常用于建立现实情境模型，如抛体运动、面积和利润。在典型题目中，需要设未知数、建立二次方程、求解，并结合题意解释答案。当变量表示物理量时，应舍去负数或不合理的解。
rises as (x + 8)(x - 5) = 0. Since length cannot be negative, x = 5 is the valid solution.
例如，如果一个矩形的宽为 x，长为 x + 3，面积为 40，则 x(x + 3) = 40。得到 x² + 3x - 40 = 0，分解为 (x + 8)(x - 5) = 0。由于长度不能为负，所以 x = 5 是有效解。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
rrors when substituting into the quadratic formula. In exams, always show your working, check your solutions by substitution, and state the nature of the roots when asked.
常见错误包括：因式分解前忘记将方程设为零；未检查 x = 0 就两边同时除以 x；代入求根公式时出现符号错误。考试中务必写出步骤，通过代入检验解，并在要求时说明根的性质。
- Set the equation to zero first | 先将方程设为零
- Do not divide by x unless x = 0 is handled separately | 除非单独处理 x = 0，否则不要两边除以 x
- Check your final answers by substitution | 通过代入检验最终答案
10. Summary Checklist | 复习清单
To master quadratic equations, make sure you can recognise the standard form, factorise simple and harder quadratics, complete the square, use the quadratic formula, interpret the discriminant, sketch the parabola, and solve application problems. Regular practice with past-paper questions is the best way to build confidence.
要掌握二次方程，请确保你能够识别标准式、分解简单和较难的二次式、配方、使用求根公式、解释判别式、绘制抛物线以及解决应用题。定期练习历年真题是建立信心的最佳方式。
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