Solving Quadratic Equations by Factorisation | 用因式分解解一元二次方程

📚 Solving Quadratic Equations by Factorisation | 用因式分解解一元二次方程

Quadratic equations appear everywhere in IGCSE Mathematics, from algebra to problem-solving questions. One of the fastest and most elegant ways to solve them is by factorisation. In this revision guide, we will break down the process step by step, using clear examples and exam-style tips.

一元二次方程在 IGCSE 数学中随处可见，从代数题到应用题都会涉及。其中最快捷、最优雅的解法之一就是因式分解。在本复习指南中，我们将一步步拆解这个过程，配合清晰的例题和考试技巧。

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1. What Is a Quadratic Equation? | 什么是一元二次方程

A quadratic equation is an equation that can be written in the general form:

ax² + bx + c = 0

where a, b and c are constants, and a ≠ 0. The highest power of x is 2, which is why it is called “quadratic”. For example, 3x² − 5x + 2 = 0 is a quadratic equation, while 2x + 1 = 0 is not.

一元二次方程是指可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。x 的最高次数为 2，因此称为“二次”。例如，3x² − 5x + 2 = 0 是一元二次方程，而 2x + 1 = 0 不是。

To solve a quadratic equation means to find the value(s) of x that make the equation true. A quadratic equation can have two real solutions, one repeated solution, or no real solutions. Factorisation helps us find these solutions when the expression factorises neatly.

解一元二次方程就是求出使方程成立的 x 的值。一元二次方程可以有两个实数解、一个重根，也可能没有实数解。当表达式可以整齐分解时，因式分解能帮助我们找到这些解。

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2. The Zero Product Property | 零乘积性质

The key idea behind solving by factorisation is the zero product property. It states that if the product of two expressions is zero, then at least one of the expressions must be zero.

因式分解解法的核心思想是“零乘积性质”：如果两个表达式的乘积为零，那么其中至少一个表达式必须为零。

If A × B = 0, then A = 0 or B = 0

For example, if (x − 3)(x + 2) = 0, then x − 3 = 0 or x + 2 = 0. This gives x = 3 or x = −2. This property works because zero is the only number that, when multiplied by anything else, still gives zero.

例如，如果 (x − 3)(x + 2) = 0，那么 x − 3 = 0 或 x + 2 = 0，解得 x = 3 或 x = −2。这个性质成立是因为只有零与任何数相乘仍为零。

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3. Factorising Quadratics with a = 1 | 因式分解二次项系数为 1 的情形

When the coefficient of x² is 1, the quadratic has the form x² + bx + c. We look for two numbers p and q such that:

当 x² 的系数为 1 时，二次式形如 x² + bx + c。我们要找两个数 p 和 q，使得：

p + q = b and p × q = c

Then we can write x² + bx + c = (x + p)(x + q). Let us factorise x² + 5x + 6. We need two numbers that add to 5 and multiply to 6. The numbers 2 and 3 work, so:

于是我们可以写成 x² + bx + c = (x + p)(x + q)。我们试着分解 x² + 5x + 6。需要找两个数相加等于 5、相乘等于 6。2 和 3 满足条件，所以：

x² + 5x + 6 = (x + 2)(x + 3)

Always check the signs carefully. For x² − 7x + 12, we need two numbers that add to −7 and multiply to 12. Both numbers must be negative, so −3 and −4 work: (x − 3)(x − 4).

要仔细检查符号。对于 x² − 7x + 12，我们需要两个数相加等于 −7、相乘等于 12。两个数都必须是负数，所以 −3 和 −4 满足：得到 (x − 3)(x − 4)。

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4. Factorising Quadratics with a ≠ 1 | 因式分解二次项系数不为 1 的情形

When the coefficient of x² is not 1, we use the “split the middle term” method. Consider ax² + bx + c:

当 x² 的系数不为 1 时，我们使用“拆中项”的方法。以 ax² + bx + c 为例：

• Step 1: Multiply a and c to get the product ac.
• 第一步：将 a 和 c 相乘得到积 ac。
• Step 2: Find two numbers that add to b and multiply to ac.
• 第二步：找两个数相加等于 b、相乘等于 ac。
• Step 3: Split the middle term bx into two terms using these numbers.
• 第三步：用这两个数把中间项 bx 拆成两项。
• Step 4: Factorise by grouping in pairs.
• 第四步：两两分组进行因式分解。

Let us factorise 2x² + 7x + 3. Here a = 2, b = 7, c = 3, so ac = 6. We need two numbers that add to 7 and multiply to 6: these are 1 and 6. Split the middle term: 2x² + x + 6x + 3. Now factorise by grouping:

我们分解 2x² + 7x + 3。这里 a = 2，b = 7，c = 3，所以 ac = 6。需要找两个数相加等于 7、相乘等于 6：这两个数是 1 和 6。拆中项得 2x² + x + 6x + 3，然后分组分解：

2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)

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5. Solving a Quadratic by Factorisation | 用因式分解求解一元二次方程

To solve a quadratic equation by factorisation, follow these steps:

用因式分解法解一元二次方程，请按以下步骤操作：

• Step 1: Rearrange the equation so that one side equals zero.
• 第一步：移项使方程一边等于零。
• Step 2: Factorise the quadratic expression completely.
• 第二步：将二次表达式完全因式分解。
• Step 3: Use the zero product property to set each factor to zero.
• 第三步：利用零乘积性质，令每个因式等于零。
• Step 4: Solve the resulting linear equations.
• 第四步：解所得的线性方程。

Example: solve x² − 5x = 14. First bring everything to one side: x² − 5x − 14 = 0. We need two numbers that add to −5 and multiply to −14: these are −7 and 2. Factorise: (x − 7)(x + 2) = 0. Therefore x − 7 = 0 or x + 2 = 0, giving x = 7 or x = −2.

例：解 x² − 5x = 14。首先把所有项移到一边，得 x² − 5x − 14 = 0。我们需要两个数相加等于 −5、相乘等于 −14：这两个数是 −7 和 2。因式分解得 (x − 7)(x + 2) = 0。因此 x − 7 = 0 或 x + 2 = 0，解得 x = 7 或 x = −2。

Let us also solve 2x² + 7x + 3 = 0 using the factorisation from Section 4:

我们再利用第 4 节的因式分解结果来解 2x² + 7x + 3 = 0：

(2x + 1)(x + 3) = 0

Set 2x + 1 = 0 → x = −½, and set x + 3 = 0 → x = −3. The solutions are x = −½ or x = −3.

令 2x + 1 = 0，得 x = −½；令 x + 3 = 0，得 x = −3。因此解为 x = −½ 或 x = −3。

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6. Special Case: Difference of Two Squares | 特殊情况：平方差公式

If a quadratic has the form x² − k, where k > 0, we can use the difference of two squares identity:

如果二次式形如 x² − k，其中 k > 0，我们可以使用平方差公式：

x² − k = (x − √k)(x + √k)

For example, solve x² − 9 = 0. Since 9 = 3², we write (x − 3)(x + 3) = 0, so x = 3 or x = −3. This explains why the equation x² = 9 has two solutions: x = ±3.

例如，解 x² − 9 = 0。因为 9 = 3²，所以写成 (x − 3)(x + 3) = 0，于是 x = 3 或 x = −3。这也说明了为什么方程 x² = 9 有两个解：x = ±3。

When solving x² = k by taking square roots, remember to include the negative root. For x² = 25, we get x = ±5. Many students lose marks by writing only x = 5.

用开平方的方法解 x² = k 时，记得包含负根。例如 x² = 25，应得 x = ±5。许多学生只写 x = 5 而丢分。

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7. Common Mistakes to Avoid | 常见错误提醒

Here are the most frequent errors that IGCSE students make when solving quadratics by factorisation:

以下是 IGCSE 学生在用因式分解解一元二次方程时最常见的错误：

Mistake | 错误	Correct Method | 正确做法
Factorising without moving all terms to one side, e.g. solving x² + 5x = −6 by factorising x² + 5x first. | 未把所有项移到一边就因式分解，例如先分解 x² + 5x 来解 x² + 5x = −6。	Rewrite as x² + 5x + 6 = 0, then factorise (x + 2)(x + 3) = 0. | 改写为 x² + 5x + 6 = 0，再分解为 (x + 2)(x + 3) = 0。
Forgetting the negative root: solving x² = 16 and writing x = 4 only. | 忘记负根：解 x² = 16 只写 x = 4。	Write x = ±4. | 应写 x = ±4。
Sign errors when factorising, such as writing (x − 2)(x + 3) for x² + x − 6. | 因式分解时符号错误，例如把 x² + x − 6 写成 (x − 2)(x + 3)。	Check by expanding: (x − 2)(x + 3) = x² + x − 6, so this is actually correct. Always expand to verify. | 通过展开检验：(x − 2)(x + 3) = x² + x − 6，所以其实是对的。务必展开验证。

Always expand your factorised answer back to check that it matches the original quadratic. This takes ten seconds and prevents careless errors.

务必把因式分解的结果展开，检查是否与原二次式一致。这只需要十秒钟，却能避免粗心错误。

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8. Practice Questions | 练习题目

Attempt these questions on your own before checking the answers below.

请先独立完成以下题目，再对照下面给出的答案。

1. Solve x² + 7x + 10 = 0.

   解方程 x² + 7x + 10 = 0。

2. Solve x² − 4x − 21 = 0.

   解方程 x² − 4x − 21 = 0。

3. Solve 3x² − 10x + 3 = 0.

   解方程 3x² − 10x + 3 = 0。

4. Solve 2x² + 5x = 12.

   解方程 2x² + 5x = 12。

5. Solve 4x² − 25 = 0.

   解方程 4x² − 25 = 0。

Question | 题号	Answer | 答案
1	x = −2 or x = −5 | x = −2 或 x = −5
2	x = 7 or x = −3 | x = 7 或 x = −3
3	x = 3 or x = ⅓ | x = 3 或 x = ⅓
4	x = 3⁄2 or x = −4 | x = 3⁄2 或 x = −4
5	x = 5⁄2 or x = −5⁄2 | x = 5⁄2 或 x = −5⁄2

For question 4, first rearrange to 2x² + 5x − 12 = 0. The factorised form is (2x − 3)(x + 4) = 0. For question 5, use the difference of two squares: (2x − 5)(2x + 5) = 0.

第 4 题先移项得 2x² + 5x − 12 = 0，分解为 (2x − 3)(x + 4) = 0。第 5 题使用平方差公式：(2x − 5)(2x + 5) = 0。

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9. Exam Tips | 考试技巧

Here are some practical tips to maximise your marks in the IGCSE exam:

以下是一些能在 IGCSE 考试中帮你拿到最多分数的实用建议：

• Always rearrange to ax² + bx + c = 0 before factorising, unless the question already gives this form.
• 在因式分解前，务必先整理成 ax² + bx + c = 0 的形式，除非题目已经给出该形式。
• Look for a common factor first. For example, 2x² − 8x = 0 becomes 2x(x − 4) = 0, giving x = 0 or x = 4. Do not divide both sides by x, because this loses the solution x = 0.
• 先找公因式。例如，2x² − 8x = 0 化为 2x(x − 4) = 0，得 x = 0 或 x = 4。切勿两边同时除以 x，因为会丢失 x = 0 这个解。
• If the quadratic does not factorise easily, use the quadratic formula or completing the square instead. Factorisation is only one method.
• 如果二次式不容易因式分解，可以改用求根公式或配方法。因式分解只是其中一种方法。
• State your final answer clearly as “x = … or x = …”. Use the word “or”, not “and”.
• 清楚地写出最终答案“x = … 或 x = …”。用“或”，不要用“和”。

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10. Summary | 总结

Solving quadratic equations by factorisation is a core skill for IGCSE Mathematics. Remember the three essential steps: rearrange to zero, factorise, then apply the zero product property. With regular practice, you will recognise common factorisation patterns quickly and accurately.

用因式分解解一元二次方程是 IGCSE 数学的核心技能。请牢记三个关键步骤：移项归零、因式分解、应用零乘积性质。通过规律练习，你就能快速准确地识别常见的因式分解模式。

The key facts from this guide are:

本指南的关键要点如下：

• A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0.
• 一元二次方程形如 ax² + bx + c = 0，其中 a ≠ 0。
• If A × B = 0, then A = 0 or B = 0.
• 若 A × B = 0，则 A = 0 或 B = 0。
• For x² + bx + c, find two numbers that add to b and multiply to c.
• 对 x² + bx + c，找两个数相加得 b、相乘得 c。
• Use the split-the-middle-term method when a ≠ 1.
• 当 a ≠ 1 时，使用拆中项法。
• Remember the difference of two squares: x² − k = (x − √k)(x + √k).
• 记住平方差公式：x² − k = (x − √k)(x + √k)。

Keep practising past-paper questions, check your answers by substitution or expansion, and you will master quadratic factorisation in no time.

坚持练习历年真题，并用代入或展开来检验答案，你很快就能掌握二次因式分解。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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