📚 Solving Quadratic Equations | 解二次方程
A quadratic equation is one of the most important topics in IGCSE Mathematics. Mastering the different methods of solving quadratics will prepare you for higher-level algebra, graphs, and problem-solving. In this revision guide, we will explore every key method step by step, with worked examples and exam tips.
二次方程是 IGCSE 数学中最重要的话题之一。掌握解二次方程的不同方法，将为你学习更高级的代数、函数图像和实际问题打下坚实基础。在本复习指南中，我们将逐步讲解每一种核心方法，并配有例题和考试技巧。
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 general form:
二次方程是可以写成以下一般形式的方程：
ax² + bx + c = 0 , where a ≠ 0
Here, a is the coefficient of x², b is the coefficient of x, and c is a constant term. The condition a ≠ 0 is essential; if a = 0, the equation becomes linear, not quadratic.
其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。条件 a ≠ 0 至关重要；如果 a = 0，方程就变成了一次方程，而不是二次方程。
For example, 2x² − 5x + 3 = 0 is quadratic, while x + 4 = 0 is linear. A quadratic equation can have two distinct real roots, one repeated root, or no real roots, depending on its discriminant.
例如，2x² − 5x + 3 = 0 是二次方程，而 x + 4 = 0 是一次方程。二次方程可能有两个不同的实数根、一个重根，或者没有实数根，这取决于它的判别式。
2. Solving by Factorisation | 因式分解法
Factorisation is often the quickest method when the quadratic has simple integer coefficients. The idea is to write the quadratic as a product of two linear binomials, then use the zero product property: if p × q = 0, then p = 0 or q = 0.
当二次方程具有简单的整数系数时，因式分解通常是最快捷的方法。其核心思想是把二次方程写成两个一次二项式的乘积，然后利用零乘积性质：如果 p × q = 0，那么 p = 0 或 q = 0。
Consider the equation x² + 5x + 6 = 0. We look for two numbers that multiply to 6 and add to 5. These numbers are 2 and 3. Therefore:
考虑方程 x² + 5x + 6 = 0。我们需要找两个数，它们相乘等于 6，相加等于 5。这两个数是 2 和 3。因此：
(x + 2)(x + 3) = 0
Applying the zero product property gives x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3. Always check your answers by substituting them back into the original equation.
应用零乘积性质可得 x + 2 = 0 或 x + 3 = 0，所以 x = −2 或 x = −3。务必把答案代回原方程进行检验。
When the coefficient of x² is not 1, such as 2x² + 7x + 3 = 0, you must consider factor pairs of both a and c. Here we factor as (2x + 1)(x + 3) = 0, giving x = −½ or x = −3.
当 x² 的系数不为 1 时，例如 2x² + 7x + 3 = 0，你需要同时考虑 a 和 c 的因数对。这里我们可以分解为 (2x + 1)(x + 3) = 0，得到 x = −½ 或 x = −3。
3. Solving by Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This method is particularly useful for solving equations that cannot be factorised and for finding the vertex of a parabola.
配方法将二次方程改写为 a(x + p)² + q 的形式。这种方法特别适用于无法因式分解的方程，也常用于求抛物线的顶点。
For the equation x² + 6x − 7 = 0, we first halve the coefficient of x: 6 ÷ 2 = 3. Write (x + 3)² and subtract 3² to keep the expression balanced:
对于方程 x² + 6x − 7 = 0，我们先把 x 的系数除以 2：6 ÷ 2 = 3。写出 (x + 3)²，并减去 3² 以保持表达式平衡：
x² + 6x − 7 = (x + 3)² − 9 − 7 = (x + 3)² − 16
Setting (x + 3)² − 16 = 0 gives (x + 3)² = 16. Taking the square root of both sides: x + 3 = ±4, so x = 1 or x = −7.
令 (x + 3)² − 16 = 0，得 (x + 3)² = 16。两边开平方：x + 3 = ±4，所以 x = 1 或 x = −7。
Remember: when taking a square root, you must include both the positive and negative solutions. This is a common source of lost marks in exams.
请记住：开平方时，必须同时取正负两个解。这是考试中常见的失分点。
4. The Quadratic Formula | 二次公式
The quadratic formula solves any quadratic equation of the form ax² + bx + c = 0. It is the most general method and works even when factorisation is difficult or impossible.
二次公式可以求解任何形式为 ax² + bx + c = 0 的二次方程。它是最通用的方法，即使因式分解困难甚至不可能时也同样适用。
x = (−b ± √(b² − 4ac)) / 2a
Let us solve 2x² − 4x − 6 = 0. Here a = 2, b = −4, and c = −6. Substitute into the formula:
让我们求解 2x² − 4x − 6 = 0。这里 a = 2，b = −4，c = −6。代入公式：
x = (4 ± √(16 + 48)) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4
This gives x = (4 + 8)/4 = 3 or x = (4 − 8)/4 = −1. So the solutions are x = 3 and x = −1.
因此 x = (4 + 8)/4 = 3 或 x = (4 − 8)/4 = −1。所以方程的解为 x = 3 和 x = −1。
Always write the values of a, b, and c before substituting to avoid careless errors. In the IGCSE exam, the formula is usually provided on the formula sheet, but you must know exactly how to use it.
在代入之前，务必先写出 a、b、c 的值，以避免粗心错误。在 IGCSE 考试中，公式表通常会给出二次公式，但你必须准确知道如何使用它。
5. The Discriminant and the Nature of Roots | 判别式与根的性质
The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells us the nature of the roots without solving the equation fully.
二次公式中 b² − 4ac 这一部分称为判别式。它可以在不完全求解方程的情况下告诉我们根的性质。
We define the discriminant as Δ = b² − 4ac. There are three cases:
我们定义判别式为 Δ = b² − 4ac。共有三种情况：
| Discriminant Δ | Nature of Roots | Graph Interpretation |
| Δ > 0 | Two distinct real roots | Parabola crosses the x-axis at two points |
| Δ = 0 | One repeated real root | Parabola touches the x-axis at one point |
| Δ < 0 | No real roots | Parabola does not intersect the x-axis |
For example, for x² − 4x + 4 = 0, the discriminant is Δ = 16 − 16 = 0, so the equation has exactly one repeated root: x = 2.
例如，对于 x² − 4x + 4 = 0，判别式 Δ = 16 − 16 = 0，因此方程只有一个重根：x = 2。
Questions often ask you to find the range of values of k for which an equation has two real roots. Set Δ > 0 and solve the resulting inequality. This is a common extended-paper question.
考试题常要求你求 k 的取值范围，使得方程有两个实数根。此时令 Δ > 0 并解对应不等式即可。这是扩展卷的常见题型。
6. Solving Quadratic Equations by Graphing | 图像法求解
Graphically, the real roots of a quadratic equation ax² + bx + c = 0 are the x-coordinates of the points where the parabola y = ax² + bx + c crosses the x-axis.
从图像上看，二次方程 ax² + bx + c = 0 的实数根，就是抛物线 y = ax² + bx + c 与 x 轴交点的横坐标。
To sketch the graph, you should identify three key features: the y-intercept (0, c), the roots or the discriminant, and the vertex. The vertex can be found by completing the square or using x = −b/(2a).
画图时需要确定三个关键特征：y 轴截距 (0, c)、根或判别式，以及顶点。顶点可以通过配方法或 x = −b/(2a) 求得。
If the discriminant is positive, the curve crosses the axis twice; if it is zero, the vertex lies exactly on the axis; if negative, the curve never touches the axis.
如果判别式为正，曲线与 x 轴交于两点；如果判别式为零，顶点恰好落在 x 轴上；如果判别式为负，曲线与 x 轴没有交点。
In IGCSE exams, you may be asked to draw a graph from a table of values and then estimate the roots. Always read the x-intercepts carefully and give your answers to the degree of accuracy requested.
在 IGCSE 考试中，你可能会被要求根据数值表作图，然后估计根。务必仔细读取 x 轴交点，并按题目要求的精度给出答案。
7. Quadratic Word Problems | 二次方程应用题
Many examination problems present quadratics in context. Common settings include area problems, number problems, and projectile motion in physics.
许多考试题目会把二次方程放在实际情景中。常见的有面积问题、数字问题，以及物理中的抛体运动。
Consider this example: a rectangle has length 4 cm longer than its width, and its area is 60 cm². Let the width be x cm. Then the length is (x + 4) cm, and the area gives:
看这个例子：一个长方形的长比宽长 4 cm，面积为 60 cm²。设宽为 x cm，则长为 (x + 4) cm，由面积可得：
x(x + 4) = 60 → x² + 4x − 60 = 0
Factorising gives (x + 10)(x − 6) = 0, so x = −10 or x = 6. Since length cannot be negative, the width is 6 cm and the length is 10 cm.
因式分解得 (x + 10)(x − 6) = 0，所以 x = −10 或 x = 6。由于长度不能为负，所以宽为 6 cm，长为 10 cm。
The key is to define your variable clearly, translate the words into an equation, solve, and then interpret the roots in the context of the problem. Reject any solution that does not make sense physically.
关键是要清晰地设定未知数，把文字翻译成方程，解方程，然后结合实际问题背景解释根。任何不符合物理意义的解都要舍去。
8. Common Mistakes and Exam Tips | 常见错误与考试技巧
Below are the most frequent errors students make when solving quadratic equations in IGCSE exams.
以下是学生在 IGCSE 考试中解二次方程时最常犯的错误。
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Forgetting to rearrange the equation into standard form. Always move all terms to one side so the equation equals zero before factorising or using the formula.
忘记将方程整理成标准形式。在使用因式分解或公式前，务必把所有项移到一边，使方程等于零。
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Dropping the ± sign when taking square roots. If (x + 3)² = 16, then x + 3 = +4 or −4. Missing the negative root loses a mark.
开平方时漏掉 ± 号。如果 (x + 3)² = 16，则 x + 3 = +4 或 −4。漏掉负根会丢分。
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Sign errors when substituting into the quadratic formula. Write down a, b, c with their signs first. For example, if b = −4, then −b = 4.
代入二次公式时出现符号错误。先把 a、b、c 连同符号写出来。例如，若 b = −4，则 −b = 4。
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Dividing both sides by x incorrectly. If x is a possible root, dividing by x loses that root. Instead, factorise x out: x(ax + b) = 0.
错误地两边同时除以 x。如果 x 可能是根，除以 x 会丢失这个根。应该提取公因式 x：x(ax + b) = 0。
To score full marks, always show your working in a logical order. In the mark scheme, method marks are awarded even if your final answer is incorrect, so never write only the answer.
要拿到满分，请按逻辑顺序展示你的解题过程。在评分标准中，即使最终答案有误，方法分也会被给予，因此绝不要只写答案。
9. Practice Questions | 巩固练习
Try these questions on your own before checking the answers. They cover all the methods discussed in this guide.
请先独立完成以下练习，再核对答案。这些题目涵盖了本指南讨论的所有方法。
Question 1: Solve x² − 9x + 20 = 0 by factorisation.
题目 1：用因式分解法解 x² − 9x + 20 = 0。
Question 2: Solve x² + 8x − 3 = 0 by completing the square. Give your answers to two decimal places.
题目 2：用配方法解 x² + 8x − 3 = 0。答案保留两位小数。
Question 3: Use the quadratic formula to solve 3x² + 5x − 2 = 0.
题目 3：用二次公式解 3x² + 5x − 2 = 0。
Question 4: Find the value of k such that kx² − 4x + 2 = 0 has exactly one repeated root.
题目 4：求 k 的值，使方程 kx² − 4x + 2 = 0 恰好有一个重根。
Answers: (1) x = 4 or x = 5. (2) x = 0.36 or x = −8.36. (3) x = ⅓ or x = −2. (4) Set Δ = 0: 16 − 8k = 0, so k = 2.
答案：(1) x = 4 或 x = 5。(2) x = 0.36 或 x = −8.36。(3) x = ⅓ 或 x = −2。(4) 令 Δ = 0：16 − 8k = 0，所以 k = 2。
By practising all four methods and reviewing your mistakes, you will build both speed and accuracy. Quadratic equations appear throughout the IGCSE syllabus, and mastering them is an investment in your final grade.
通过练习这四种方法并认真复习错误，你将同时提升速度和准确率。二次方程贯穿整个 IGCSE 课程大纲，掌握它是对你最终成绩的重要投资。
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