📚 Quadratic Equations | 二次方程
Welcome to this IGCSE Mathematics revision guide on quadratic equations. In this article, we will explore the standard form of a quadratic equation, the main methods for solving it, the role of the discriminant, and how quadratic functions appear graphically and in real-life problems. Understanding these ideas is essential for success in the IGCSE syllabus.
欢迎阅读本篇 IGCSE 数学复习指南，主题是二次方程。本文将探讨二次方程的标准形式、主要求解方法、判别式的作用，以及二次函数在图像和实际生活中的应用。掌握这些内容对在 IGCSE 考试中取得好成绩至关重要。
1. What is a Quadratic Equation? | 什么是二次方程？
n becomes linear, not quadratic. The highest power of the variable x is 2, which is why it is called 'quadratic'.
二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数，且 a ≠ 0。a ≠ 0 这一条件是必要的，因为如果 a = 0，方程就变成了一次方程，而不是二次方程。变量 x 的最高次数是 2，因此它被称为 '二次' 方程。
uations can have two real solutions, one repeated solution, or no real solutions, depending on the values of the coefficients.
常见的例子包括 x² - 5x + 6 = 0、2x² + 3x - 2 = 0 和 x² - 4 = 0。根据系数的取值，二次方程可以有两个实数解、一个重根或没有实数解。
2. Solving by Factorisation | 因式分解法
0. Using the zero product property, either x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.
当二次式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。例如，x² - 5x + 6 可以写成 (x - 2)(x - 3)，因此方程 x² - 5x + 6 = 0 变为 (x - 2)(x - 3) = 0。根据零乘积性质，x - 2 = 0 或 x - 3 = 0，得到 x = 2 或 x = 3。
rite it as x² + 3x - 10 = 0, then factorise to (x + 5)(x - 2) = 0, giving x = -5 or x = 2.
在因式分解之前，一定要确保方程右边等于零。例如，要解 x² + 3x = 10，应先将它改写为 x² + 3x - 10 = 0，然后因式分解为 (x + 5)(x - 2) = 0，得到 x = -5 或 x = 2。
3. Completing the Square | 配方法
9 = 16, which is (x + 3)² = 16. Taking square roots gives x + 3 = ±4, so x = 1 or x = -7.
配方法将二次方程转化为 (x + p)² = q 的形式，然后通过开平方来求解。例如，x² + 6x - 7 = 0 可以改写为 x² + 6x = 7。两边加上 (6/2)² = 9，得到 x² + 6x + 9 = 16，即 (x + 3)² = 16。开平方得到 x + 3 = ±4，所以 x = 1 或 x = -7。
This method is especially useful for deriving the quadratic formula and for finding the coordinates of the vertex of a parabola. It also works when the coefficient of x² is not 1, but you must first divide the whole equation by that coefficient.
这种方法在推导求根公式以及求抛物线的顶点坐标时特别有用。当 x² 的系数不是 1 时，配方法同样有效，但你必须先把整个方程除以该系数。
4. The Quadratic Formula | 求根公式
The quadratic formula gives the solutions of ax² + bx + c = 0 directly:
求根公式直接给出 ax² + bx + c = 0 的解：
x = (-b ± √b² - 4ac) / 2a
2, b = 3 and c = -2. Substituting gives x = (-3 ± √(3² - 4 × 2 × (-2))) / (2 × 2) = (-3 ± √25) / 4 = (-3 ± 5) / 4, so x = 1/2 or x = -2.
这个公式适用于所有二次方程，即使在因式分解困难或无法进行时也能使用。例如，对于 2x² + 3x - 2 = 0，我们有 a = 2、b = 3、c = -2。代入公式得到 x = (-3 ± √(3² - 4 × 2 × (-2))) / (2 × 2) = (-3 ± √25) / 4 = (-3 ± 5) / 4，所以 x = 1/2 或 x = -2。
Make sure to write the formula with the entire numerator divided by 2a, and use brackets carefully when substituting negative values. A common error is to apply the negative sign incorrectly to -b.
请确保公式的整个分子都除以 2a，并在代入负值时小心使用括号。一个常见错误是把 -b 的负号处理错误。
5. The Discriminant | 判别式
is one repeated real root. If D < 0, there are no real roots, but two complex roots exist.
二次方程的判别式是表达式 D = b² - 4ac。它可以在不完整解方程的情况下告诉我们根的性质。如果 D > 0，则有两个不同的实数根；如果 D = 0，则有一个重复的实数根；如果 D < 0，则没有实数根，但存在两个复数根。
The table below summarises the relationship between the discriminant and the number of real solutions.
下表总结了判别式与实数解个数之间的关系。
| Discriminant D = b² - 4ac | Number of real roots | Example |
|---|---|---|
| D > 0 | 2 distinct real roots | x² - 5x + 6 = 0, D = 1 |
| D = 0 | 1 repeated real root | x² - 6x + 9 = 0, D = 0 |
| D < 0 | No real roots | x² + 2x + 5 = 0, D = -16 |
uation has, or to find values of k for which a quadratic has equal roots by setting D = 0.
在 IGCSE 考试中，你可能会被要求判断一个方程有多少个根，或者通过令 D = 0 来求使二次方程有等根的 k 值。
6. Graphs of Quadratic Functions | 二次函数图像
on ax² + bx + c = 0 are the x-intercepts of the graph, where the curve crosses the x-axis.
函数 y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，有一个最低点；如果 a < 0，开口向下，有一个最高点。方程 ax² + bx + c = 0 的根就是图像与 x 轴交点的横坐标。
back into the function gives the y-coordinate of the vertex. The y-intercept is at (0, c).
抛物线的顶点可以通过配方法或使用 x = -b / 2a 来求得。将这个 x 值代回函数就可以得到顶点的纵坐标。y 轴截距为 (0, c)。
Sketching quadratic graphs is a common exam task. You should be able to label the y-intercept, the x-intercepts or roots, and the coordinates of the vertex.
绘制二次函数图像是常见的考试题目。你应该能够标出 y 轴截距、x 轴截距或根，以及顶点的坐标。
7. Applications and Word Problems | 实际应用与文字题
Quadratic equations often appear in word problems involving areas, projectile motion, profit maximisation and number puzzles. The key steps are to define a variable, form an equation from the given information, solve the quadratic, and then interpret the answer in the context of the problem.
二次方程经常出现在涉及面积、抛体运动、利润最大化和数字谜题的文字题中。关键步骤是设定一个变量，根据已知信息列出方程，解这个二次方程，然后结合题意解释答案。
= 5 or x = -8. Since a width cannot be negative, the width is 5 cm and the length is 8 cm.
例如，一个长方形的长比宽多 3 厘米，面积为 40 平方厘米。设宽为 x 厘米，则长为 (x + 3) 厘米。面积方程为 x(x + 3) = 40，即 x² + 3x - 40 = 0。因式分解得到 (x + 8)(x - 5) = 0，所以 x = 5 或 x = -8。由于宽度不能为负，因此宽为 5 厘米，长为 8 厘米。
Always reject solutions that do not make sense in real life, such as negative lengths, negative time or fractional answers when whole objects are required.
一定要舍去在实际生活中没有意义的解，例如负的长度、负的时间，或者在需要整数对象时出现的小数答案。
8. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many students lose marks by forgetting that the coefficient a must not be zero, by dropping a negative sign when substituting into the formula, or by not setting the equation equal to zero before factorising. Another frequent error is only writing down one solution when there should be two, especially when taking a square root.
许多学生因为忘记系数 a 不能为零、代入公式时丢掉负号、或因式分解前没有让方程右边等于零而失分。另一个常见错误是只写出一个解，而实际上应该有两个解，特别是在开平方时更容易遗漏。
To avoid these mistakes, always write out the values of a, b and c clearly, pay attention to brackets and signs, and check your solutions by substituting them back into the original equation. If you have time, sketch the graph to see whether the number of solutions matches your answer.
为了避免这些错误，请始终清楚地写出 a、b 和 c 的值，注意括号和符号，并通过将解代回原方程来检查。如果时间允许，画出函数图像，看看解的个数是否与你的答案相符。
In the IGCSE exam, you should be able to recognise which method is most efficient: use factorisation for simple quadratics, complete the square when asked to find the vertex, and use the quadratic formula for messy or irrational solutions. Practising mixed exercises will help you choose the best approach quickly.
在 IGCSE 考试中，你应该能够判断哪种方法最有效：简单的二次方程用因式分解，要求求顶点时用配方法，对于较复杂或无理数解使用求根公式。多做混合练习能帮助你快速选择最佳方法。
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