📚 Solving Quadratic Equations | 解二次方程
Quadratic equations are one of the most important topics in the IGCSE Edexcel Mathematics syllabus. They appear in algebra, graphs, geometry, and problem-solving questions. Mastering this topic will significantly improve your exam performance.
二次方程是 IGCSE Edexcel 数学考纲中最重要的内容之一。它出现在代数、函数图像、几何和应用题中。掌握这一主题将显著提升你的考试成绩。
1. Standard Form and Key Concepts | 标准形式与核心概念
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers, and a ≠ 0.
二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为实数，且 a ≠ 0。
The highest power of the variable is 2, which is why it is called 'quadratic'. The coefficient 'a' cannot be zero, or else the equation becomes linear.
变量的最高次数是 2，因此被称为“二次”。系数 a 不能为零，否则方程就变成了线性方程。
2. Solving by Factorisation | 因式分解法
Factorisation is the quickest method when the quadratic can be written as a product of two linear factors. The idea is: if p × q = 0, then either p = 0 or q = 0.
当二次式可以写成两个一次因式的乘积时，因式分解是最快的方法。核心思想是：如果 p × q = 0，则 p = 0 或 q = 0。
ply to 6 and add to -5. These are -2 and -3. So (x - 2)(x - 3) = 0, giving x = 2 or x = 3.
例如：解 x² - 5x + 6 = 0。需要找到两个数，乘积为 6，和为 -5。这两个数是 -2 和 -3。所以 (x - 2)(x - 3) = 0，得到 x = 2 或 x = 3。
Steps for factorisation:
因式分解步骤：
- Write the equation in the form ax² + bx + c = 0.
- Find two numbers that multiply to ac and add to b.
- Rewrite the middle term and factor by grouping.
- Set each bracket equal to zero and solve.
- 将方程写成 ax² + bx + c = 0 的形式。
- 找到两个数，它们的乘积等于 ac，和等于 b。
- 重写中间项并进行分组分解。
- 令每个括号等于零，然后求解。
0, then factor: 2x(x + 2) + 1(x + 2) = 0, so (2x + 1)(x + 2) = 0, giving x = -½ or x = -2.
当系数 a ≠ 1 时，方法仍然适用，但需要更细心。例如，2x² + 5x + 2 = 0。这里 ac = 4，两个数为 4 和 1。重写：2x² + 4x + x + 2 = 0，然后分解：2x(x + 2) + 1(x + 2) = 0，所以 (2x + 1)(x + 2) = 0，得到 x = -½ 或 x = -2。
3. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This method is useful for solving equations that do not factorise easily, and it also helps in graphing and finding maximum or minimum values.
配方法将二次式改写为 a(x + p)² + q 的形式。这种方法适用于不易因式分解的方程，同时也有助于作图和求最大值或最小值。
Example: Solve x² + 6x - 1 = 0 by completing the square.
例如：用配方法解 x² + 6x - 1 = 0。
is 3, and square it to get 9. Write (x + 3)² - 9 - 1 = 0, so (x + 3)² = 10. Then x + 3 = ±√10, so x = -3 ± √10.
先看 x² + 6x。取 6 的一半为 3，平方得 9。写成 (x + 3)² - 9 - 1 = 0，即 (x + 3)² = 10。于是 x + 3 = ±√10，所以 x = -3 ± √10。
The general rule: For x² + bx, add and subtract (b/2)². If the coefficient a is not 1, factor it out first before completing the square.
一般规则：对于 x² + bx，加上并减去 (b/2)²。如果系数 a 不为 1，则先提取 a 再配方。
4. The Quadratic Formula | 求根公式
solves any quadratic equation directly. For ax² + bx + c = 0, the solutions are given by:
求根公式可以直接解任何二次方程。对于 ax² + bx + c = 0，解为：
x = (-b ± √b² - 4ac) / (2a)
This formula is essential for equations that cannot be factorised easily or when completing the square becomes too lengthy.
这个公式对于无法轻易因式分解或配方过于繁琐的方程至关重要。
Example: Solve 2x² - 4x - 3 = 0 using the quadratic formula.
例如：用求根公式解 2x² - 4x - 3 = 0。
Here a = 2, b = -4, c = -3. Substitute into the formula:
这里 a = 2，b = -4，c = -3。代入公式：
x = (4 ± √((-4)² - 4×2×(-3))) / (2×2) = (4 ± √16 + 24) / 4 = (4 ± √40) / 4
Simplify √40 = 2√10, so x = (4 ± 2√10)/4 = 1 ± √10/2.
化简 √40 = 2√10，所以 x = (4 ± 2√10)/4 = 1 ± √10/2。
5. The Discriminant and 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 how many real roots a quadratic equation has without solving it completely.
求根公式中的 b² - 4ac 称为判别式。它无需完全解方程就能告诉我们二次方程实根的个数。
| Discriminant | Nature of roots | Graph interpretation |
| b² - 4ac > 0 | Two distinct real roots | Parabola crosses the x-axis at two points |
| b² - 4ac = 0 | One repeated real root | Parabola touches the x-axis at one point (vertex on axis) |
| b² - 4ac < 0 | No real roots | Parabola does not intersect the x-axis |
判别式大于零时，方程有两个不同的实数根；等于零时，有一个重复的实数根；小于零时，没有实数根。
If the discriminant is a perfect square, the quadratic factorises over integers. If it is positive but not a perfect square, the roots are irrational.
如果判别式是一个完全平方数，则二次式可以在整数范围内因式分解。如果它为正但不是完全平方数，则根为无理数。
6. Solving by Graphing | 图像法求解
to the equation ax² + bx + c = 0 are the x-coordinates where the curve crosses the x-axis.
图像法是指画出抛物线 y = ax² + bx + c。方程 ax² + bx + c = 0 的解就是曲线与 x 轴交点的横坐标。
If the graph does not cross the x-axis, the equation has no real roots. If it just touches the x-axis, there is one repeated root. This visual interpretation is often tested in exam questions.
如果图像不与 x 轴相交，则方程没有实数根。如果刚好相切，则有一个重根。这种图像解释经常在考试题目中出现。
In the iGCSE Edexcel exam, you may be given part of a graph and asked to read off approximate roots. You may also need to solve simultaneous equations graphically, including a line and a quadratic.
在 Edexcel IGCSE 考试中，你可能会得到部分图形并要求读出近似根。还可能要求通过画图来解联立方程，包括一条直线和一条抛物线。
7. Word Problems Involving Quadratics | 二次方程应用题
Applying quadratic equations to real-life situations is a common exam skill. Typical problems involve areas, projectile motion, number puzzles, and economic models.
将二次方程应用于实际生活是常见的考试能力。典型问题涉及面积、抛体运动、数字谜题和经济模型。
Example: A rectangle has length (x + 4) cm and width (x - 1) cm. Its area is 36 cm². Find x.
例如：一个长方形的长为 (x + 4) cm，宽为 (x - 1) cm。面积为 36 cm²。求 x。
rise: (x + 8)(x - 5) = 0. The positive solution is x = 5, since length cannot be negative.
列方程：(x + 4)(x - 1) = 36。展开：x² + 3x - 4 = 36，即 x² + 3x - 40 = 0。因式分解：(x + 8)(x - 5) = 0。正解为 x = 5，因为长度不能为负。
Always check whether your solutions make sense in the context. Negative lengths, times, or quantities are usually rejected.
始终检查解在情境中是否有意义。负数长度、时间或数量通常应舍去。
8. Quadratic Simultaneous Equations | 含二次的联立方程
In IGCSE Edexcel, you may need to solve a system where one equation is linear and the other is quadratic. The substitution method is the standard approach.
在 Edexcel IGCSE 中，你可能需要解一个线性方程和一个二次方程组成的方程组。代入法是标准方法。
Example: Solve y = x + 1 and y = x² - 3x + 4.
例如：解 y = x + 1 和 y = x² - 3x + 4。
+ 3 = 0. Factorise: (x - 1)(x - 3) = 0, so x = 1 or x = 3. Then substitute back to find y.
将第一个方程代入第二个：x + 1 = x² - 3x + 4。整理得 x² - 4x + 3 = 0。因式分解：(x - 1)(x - 3) = 0，所以 x = 1 或 x = 3。再代回求 y。
When x = 1, y = 2; when x = 3, y = 4. The solutions are (1, 2) and (3, 4).
当 x = 1 时，y = 2；当 x = 3 时，y = 4。解为 (1, 2) 和 (3, 4)。
These solutions represent the intersection points between the line and the parabola.
这些解代表直线与抛物线的交点。
9. Common Mistakes and How to Avoid Them | 常见错误与避坑指南
Students often make similar errors when solving quadratics. Recognising these will help you avoid losing easy marks.
学生在解二次方程时常犯类似的错误。识别这些错误能帮助你避免丢失容易得到的分数。
- Forgetting to set the equation to zero before factorising.
- Sign errors when substituting into the quadratic formula.
- Dropping the ± sign when taking square roots.
- Incorrectly expanding brackets when multiplying out.
- Not simplifying final answers (e.g., leaving fractions or surds unsimplified).
- 因式分解前忘记将方程化为等于零。
- 代入求根公式时出现符号错误。
- 取平方根时漏掉 ± 符号。
- 展开括号时错误。
- 没有化简最终答案（例如分数或根式未化简）。
Always check your solutions by substituting them back into the original equation.
始终通过将解代回原方程来检验答案。
10. Exam Tips and Summary | 考试技巧与总结
In the exam, always read the question carefully to see which method is expected. If factorisation works, use it; otherwise, the quadratic formula is a reliable backup.
考试时，仔细阅读题目以确定期望的方法。如果可以用因式分解，就用它；否则，求根公式是可靠的备选方案。
Memorise the quadratic formula and the discriminant condition. Practise completing the square, as it is often needed for solving and for finding turning points.
牢记求根公式和判别式的条件。练习配方，因为它常用于求解和找顶点。
Here is a quick reference for methods:
以下为方法速查表：
| Method | When to use |
| Factorisation | When the quadratic is factorable (discriminant is a perfect square) |
| Completing the square | When solving or finding the vertex; when a leading coefficient is 1 |
| Quadratic formula | Always works; best when no obvious factors |
| Graphing | When approximate roots are required or for verifying answers |
因式分解法适用于可分解的二次式；配方法适用于求解和求顶点；求根公式总是有效，特别在无明显因子时；图像法用于求近似根或检验答案。
With consistent practice, solving quadratic equations will become intuitive and you will gain valuable marks in your IGCSE exam.
通过持续练习，解二次方程将变得得心应手，你也将在 IGCSE 考试中获得宝贵分数。
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