Solving Quadratic Equations | 解二次方程

📚 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。

Example: Solve x² – 5x + 6 = 0. We need two numbers that multiply 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。

If the coefficient a ≠ 1, the method still works but requires extra care. For example, 2x² + 5x + 2 = 0. Here ac = 4, and the two numbers are 4 and 1. Rewrite: 2x² + 4x + x + 2 = 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。

Start with x² + 6x. Take half of 6, which 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

The quadratic formula 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

Graphical methods involve sketching or plotting the parabola y = ax² + bx + c. The solutions 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。

Set up the equation: (x + 4)(x – 1) = 36. Expand: x² + 3x – 4 = 36, so x² + 3x – 40 = 0. Factorise: (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。

Substitute the first equation into the second: x + 1 = x² – 3x + 4. Rearrange to get x² – 4x + 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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Solving Quadratic Equations | 解二次方程