📚 Solving Quadratic Equations by Factorisation | 解二次方程的因式分解法
Quadratic equations appear throughout the IGCSE Edexcel syllabus, from pure algebra questions to geometry problems involving area and projectile motion. Factorisation is often the quickest method for solving them, yet many students overlook the simple logic behind it. In this article we will explore what makes an equation quadratic, how factorisation works step by step, and how to apply it confidently under exam conditions.
从纯代数题到涉及面积和抛体运动的几何问题，二次方程在 IGCSE Edexcel 考纲中无处不在。因式分解通常是解二次方程最快的方法，但很多学生忽视了它背后简单的逻辑。本文将探究什么使一个方程成为二次方程、如何进行逐步因式分解，以及如何在考试环境中自信地应用这一方法。
1. What is a Quadratic Equation? | 什么是二次方程？
a quadratic function is a parabola. If a = 0, the equation becomes linear, not quadratic.
二次方程是指任何能够写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。未知数 x 的最高次幂为 2，这就是二次函数图像为抛物线的原因。如果 a = 0，方程就变成线性方程，而不是二次方程了。
You will encounter quadratic equations when solving problems involving areas of rectangles, right‑angled triangles (via Pythagoras), motion under gravity, and more. Recognising the structure immediately will save you time and help you choose the most efficient solving technique.
你在解决涉及矩形面积、直角三角形（用勾股定理）、重力作用下的运动等问题时都会遇到二次方程。立即识别出这种结构能节省时间并帮助你选择最有效的解法。
2. The Standard Form and Key Terminology | 标准形式与关键术语
, bx is the linear term, and c is the constant term. The coefficient a is not zero, while b and c may be zero. When b or c is missing, factorisation becomes simpler, but the equation is still quadratic.
我们总是先把方程整理成标准形式：ax² + bx + c = 0。ax² 这一项是二次项，bx 是一次项，c 是常数项。系数 a 不为零，而 b 和 c 可以为零。当缺少 b 或 c 时，因式分解会更简单，但方程仍然是二次方程。
e both quadratic even though one lacks a constant term and the other lacks a linear term. In factorisation questions on the Edexcel IGCSE, you are often expected to rearrange first.
例如，3x² − 12x = 0 和 x² − 9 = 0 都是二次方程，尽管一个缺少常数项，另一个缺少一次项。在 Edexcel IGCSE 的因式分解题目中，经常要求你先进行整理。
3. Why Factorisation Works | 为什么因式分解有效
Factorisation relies on the Zero Product Property: if the product of two expressions equals zero, then at least one of the expressions must be zero. This property is extremely powerful because it converts a single quadratic equation into two simple linear equations.
因式分解依赖于零乘积性质：如果两个式子的乘积等于零，那么至少有一个式子必须为零。这个性质非常强大，因为它把单个二次方程转化为两个简单的一次方程。
ly set each bracket equal to zero and solve. This avoids the need for trial and error with the original quadratic expression and guarantees finding all real solutions.
一旦你把二次方程写成 (px + q)(rx + s) = 0 的形式，你就可以立即令每个括号等于零并求解。这避免了在原来的二次表达式中反复试错，并能保证找到所有实数解。
4. Factorising Quadratics with a = 1 | 首项系数为 1 时的因式分解
For x² + bx + c = 0, you look for two integers m and n such that m × n = c and m + n = b.
当 x² 的系数为 1 时，因式分解通常就是寻找两个数，使它们相乘等于 c，相加等于 b。对于 x² + bx + c = 0，你需要找到两个整数 m 和 n，满足 m × n = c 且 m + n = b。
d to 7, so the factorised form is (x + 3)(x + 4) = 0. The solutions are x = −3 and x = −4.
例如，x² + 7x + 12 = 0：数字 3 和 4 的积为 12，和为 7，所以因式分解形式为 (x + 3)(x + 4) = 0。解为 x = −3 和 x = −4。
Be careful with negative signs. If c is negative, one number must be positive and the other negative. If b is negative while c is positive, both numbers will be negative, because a negative times a negative gives a positive, and their sum is negative.
要当心负号。如果 c 是负数，那么一个数为正、另一个数为负。如果 b 是负数而 c 是正数，那么两个数都是负数，因为负负得正，且它们的和为负。
5. Factorising When a > 1 | 首项系数大于 1 时的因式分解
tiply to give a×c and add to give b. Then split the middle term and factorise by grouping.
如果 x² 的系数不是 1，你还需要考虑乘积 a × c。找到两个数，使它们相乘等于 a×c，相加等于 b。然后拆分中间项，再用分组法进行因式分解。
3, then group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). Equals zero gives x = −½ or x = −3.
对于 2x² + 7x + 3，a×c = 2×3 = 6。数字 6 和 1 相乘得 6、相加得 7。改写为 2x² + 6x + x + 3，然后分组：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。令其等于零得 x = −½ 或 x = −3。
Edexcel IGCSE often includes questions where a is a small integer greater than 1, so practising the split‑and‑group method is essential. You may also spot factors directly if you are comfortable with mental arithmetic.
Edexcel IGCSE 经常出现 a 是大于 1 的小整数的题目，所以练习拆分再分组的方法至关重要。如果你心算熟练，也可以直接看出因式。
6. Difference of Two Squares | 平方差公式
− d² = 0, you can factorise instantly without searching for two numbers that sum to zero.
因式分解的一个特例是平方差公式：a² − b² = (a + b)(a − b)。当二次方程具有 x² − d² = 0 的形式时，你可以立即分解因式，而不必寻找两个和为零的数。
ents mistakenly try to find numbers that multiply to −25 and add to 0, but the direct application of the formula is faster and safer.
例如，x² − 25 = 0 可化为 (x + 5)(x − 5) = 0，得 x = −5 或 x = 5。很多学生错误地试图寻找相乘为 −25、相加为 0 的两个数，但直接套用公式更快更稳妥。
7. Solving After Factorising: the Zero Product Property | 因式分解后求解：零乘积性质
ogic applies regardless of whether the factors are linear or contain a leading coefficient.
令每个因式等于零是将因式分解转化为解的关键步骤。如果 (x − p)(x − q) = 0，那么 x − p = 0 或 x − q = 0，因此 x = p 或 x = q。无论因式是一次式还是含有首项系数，这个逻辑都适用。
Never divide both sides of a quadratic equation by an expression containing x, as you risk losing a solution. Always bring all terms to one side, factorise, and then use the zero product property.
永远不要将二次方程两边同时除以含有 x 的式子，因为这可能会丢失一个解。始终把所有项移到一边，因式分解，然后再使用零乘积性质。
8. Checking Your Solutions by Substitution | 通过代入检验解
Substitution is the most reliable way to verify that your solutions are correct. Take each value and plug it back into the original equation. If both satisfy the equation, your factorisation and solving steps were accurate.
代入法是检验你的解是否正确的最可靠方式。把每一个值代回原方程。如果两个值都能使方程成立，说明你的因式分解和求解步骤是准确的。
− 2 = 0. Checking x = −⅓: 3(1/9) − 5(−⅓) − 2 = ⅓ + 5/3 − 2 = 2 − 2 = 0. Both are correct.
例如，3x² − 5x − 2 = 0 因式分解为 (3x + 1)(x − 2) = 0，解为 x = −⅓ 和 x = 2。检验 x = 2：3(4) − 5(2) − 2 = 12 − 10 − 2 = 0。检验 x = −⅓：3(1/9) − 5(−⅓) − 2 = ⅓ + 5/3 − 2 = 2 − 2 = 0。两者都正确。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
Many errors arise from rushing the factorisation step. Forgetting to set the equation to zero first, trying to factorise when a common factor has not been taken out, and mishandling negative signs are three of the most frequent pitfalls.
许多错误源自因式分解步骤中的匆忙。忘记先将方程设为零、在未提出公因子时尝试因式分解，以及错误处理负号，是最常见的三个陷阱。
Missing a common factor complicates the factorisation and may lead to arithmetic mistakes.
在尝试对二次式进行因式分解之前，始终先检查是否存在公因子。对于 2x² − 8 = 0，提出因子 2 得到 2(x² − 4) = 0，然后再用平方差公式。直接除以 2 得到 x² − 4 = 0。漏掉公因子会使因式分解变复杂，还可能导致计算错误。
10. Applying Factorisation to Word Problems | 因式分解在应用题中的应用
IGCSE Edexcel exam questions frequently embed quadratic equations in contexts such as geometry, consecutive integer problems, and area calculations. The key is to form the equation correctly before attempting factorisation.
IGCSE Edexcel 的考题经常将二次方程嵌入几何、连续整数问题和面积计算等情境中。关键是在尝试因式分解之前先正确列出方程。
Example: "The product of two consecutive positive integers is 56. Find the integers." x − 7)(x + 8) = 0, so x = 7 or x = −8. Since integers are positive, the answer is 7 and 8.
举例：“两个连续正整数的乘积为 56，求这两个数。”设较小的为 x，则下一个为 x + 1。方程是 x(x + 1) = 56 → x² + x − 56 = 0。因式分解得 (x − 7)(x + 8) = 0，所以 x = 7 或 x = −8。由于整数是正的，答案为 7 和 8。
11. Building Speed and Confidence | 提高速度与信心
1, and then include special forms like difference of squares and common factor extraction.
由于因式分解是一项基础技能，对各种系数和符号的常规练习可以使这一过程成为自觉行为。从 a = 1 开始，再练习 a > 1 的情况，然后纳入平方差和提取公因子等特殊形式。
Use past papers from the Edexcel IGCSE Mathematics A (4MA1) specification. Questions testing factorisation often appear both in the non‑calculator and calculator papers, and being fluent will free up time for more complex problems later in the exam.
使用 Edexcel IGCSE 数学 A (4MA1) 大纲的历年真题进行练习。考查因式分解的题目常常同时出现在非计算器和计算器试卷中，熟练解题可以为你腾出时间处理后面更复杂的问题。
12. Summary and Key Takeaways | 总结与关键要点
, look for a common factor, then factorise the quadratic trinomial or recognise a special form, set each factor to zero using the zero product property, and finally check your solutions by substitution. Keep a close eye on signs throughout the process.
用因式分解法解二次方程：先整理成 ax² + bx + c = 0，寻找公因子，然后对二次三项式进行因式分解或识别特殊形式，使用零乘积性质令每个因式等于零，最后通过代入检验解。整个过程要密切留意正负号。
- Always write the equation in standard form first.
- Check for a common factor before factorising the quadratic.
- Use the zero product property – never divide by x.
- Verify answers by substituting them into the original equation.
- 始终先写成标准形式。
- 在因式分解二次式之前检查公因子。
- 使用零乘积性质——切勿除以 x。
- 将答案代入原方程进行验证。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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