📚 Solving Quadratic Equations by Factorisation | 因式分解法解二次方程
Quadratic equations appear in almost every IGCSE Mathematics paper. When the expression can be factorised, the factorisation method is the quickest and most reliable way to find the roots. In this guide, we will review the zero product property, practise factorising quadratics, and work through exam-style problems step by step.
二次方程几乎出现在每一份 IGCSE 数学试卷中。当表达式可以因式分解时，因式分解法是最快捷、最可靠的求根方法。在本指南中，我们将回顾零乘积性质，练习二次式的因式分解，并逐步解答考试风格的题目。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation in which the highest power of the unknown is 2. Its general form is:
二次方程是未知数最高次数为 2 的方程。它的一般形式为：
ax² + bx + c = 0, where a ≠ 0
Here, a, b and c are constants, and a cannot be zero because the equation would then become linear. For example, x² − 5x + 6 = 0, 2x² + 3x − 2 = 0 and x² = 9 are all quadratic equations.
其中 a、b、c 为常数，且 a 不能为零，否则方程就会变成一次方程。例如，x² − 5x + 6 = 0、2x² + 3x − 2 = 0 和 x² = 9 都是二次方程。
2. The Zero Product Property | 零乘积性质
The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In symbols:
零乘积性质指出：如果两个或多个因式的乘积为零，那么至少有一个因式必须为零。用符号表示：
If A × B = 0, then A = 0 or B = 0.
This property is the reason why factorisation works as a method for solving quadratics. If we can rewrite the equation as a product of two brackets equal to zero, we can split it into two simple linear equations and solve each one separately.
这个性质是因式分解法能求解二次方程的根本原因。如果我们能把方程改写为两个括号的乘积等于零，就可以把它拆成两个简单的一次方程，分别求解即可。
3. Review of Expanding Brackets | 复习展开括号
Before factorising, we must be comfortable expanding brackets. To expand (x + 2)(x + 3), we multiply each term in the first bracket by each term in the second bracket:
在因式分解之前，我们必须熟练展开括号。展开 (x + 2)(x + 3) 时，用第一个括号中的每一项乘以第二个括号中的每一项：
(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6
Notice that the coefficient of x is 5, which is 2 + 3, and the constant term is 6, which is 2 × 3. This pattern is exactly what we reverse when factorising quadratics.
注意 x 的系数是 5，即 2 + 3；常数项是 6，即 2 × 3。这个规律正是我们因式分解二次式时反过来使用的规律。
4. Factorising x² + bx + c | 因式分解 x² + bx + c
For quadratics of the form x² + bx + c, where the coefficient of x² is 1, we look for two numbers whose product is c and whose sum is b. Factorise x² + 5x + 6: we need two numbers that multiply to 6 and add to 5; these are 2 and 3. Therefore:
对于形如 x² + bx + c 的二次式，其中 x² 的系数为 1，我们要找两个数：它们的乘积为 c，它们的和为 b。因式分解 x² + 5x + 6：需要两个数相乘得 6、相加得 5，这两个数是 2 和 3。因此：
x² + 5x + 6 = (x + 2)(x + 3)
Watch out for negative signs. To factorise x² − x − 6, we need two numbers whose product is −6 and whose sum is −1; those numbers are 2 and −3. Hence x² − x − 6 = (x + 2)(x − 3). List the factor pairs of the constant term if you are unsure.
注意负号。因式分解 x² − x − 6 时，需要两个数乘积为 −6、和为 −1；这两个数是 2 和 −3。因此 x² − x − 6 = (x + 2)(x − 3)。如果不确定，可以列出常数项的所有因子对。
5. Factorising ax² + bx + c | 因式分解 ax² + bx + c
When a ≠ 1, we use the ‘multiply ac’ method. Take 2x² + 7x + 3 as an example. Step 1: multiply a and c, so 2 × 3 = 6. Step 2: find two numbers whose product is 6 and whose sum is b = 7; these are 1 and 6. Step 3: rewrite the middle term using these numbers and factor by grouping:
当 a ≠ 1 时，我们使用”ac 相乘”法。以 2x² + 7x + 3 为例。第 1 步：a 与 c 相乘，得 2 × 3 = 6。第 2 步：找两个数，乘积为 6、和为 b = 7，这两个数是 1 和 6。第 3 步：用这两个数重写中间项，然后分组因式分解：
2x² + 7x + 3 = 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)
Always check your factorisation by expanding the brackets. This step catches most arithmetic errors.
完成分解后，务必通过展开括号来检验结果。这一步能发现大多数计算错误。
6. The Difference of Two Squares | 平方差公式
A special and very common case is the difference of two squares. For any values a and b:
一个特殊且非常常见的情形是平方差公式。对任意 a 和 b：
a² − b² = (a + b)(a − b)
For example, x² − 25 = (x + 5)(x − 5), and 4x² − 9 = (2x)² − 3² = (2x + 3)(2x − 3). This pattern appears frequently in both factorising and solving questions.
例如，x² − 25 = (x + 5)(x − 5)，而 4x² − 9 = (2x)² − 3² = (2x + 3)(2x − 3)。这种形式在因式分解和求解题目中经常出现。
7. Solving a Quadratic Equation by Factorisation | 用因式分解法解二次方程
To solve a quadratic equation by factorisation, follow these four steps:
用因式分解法解二次方程，请按以下四个步骤操作：
- Step 1: Rearrange the equation so that one side equals zero.
第 1 步：整理方程，使一边等于零。 - Step 2: Factorise the quadratic expression completely.
第 2 步：将二次式完全因式分解。 - Step 3: Use the zero product property to set each bracket equal to zero.
第 3 步：利用零乘积性质，令每个括号等于零。 - Step 4: Solve the resulting linear equations.
第 4 步：解所得的一次方程。
Worked example: solve x² − 5x + 6 = 0. Factorising gives (x − 2)(x − 3) = 0. Therefore x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.
例题：解方程 x² − 5x + 6 = 0。因式分解得 (x − 2)(x − 3) = 0。因此 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。
Another example: solve 2x² − 5x − 3 = 0. Here a × c = −6; two numbers with product −6 and sum −5 are −6 and 1. Rewriting gives 2x² − 6x + x − 3 = 2x(x − 3) + 1(x − 3) = (2x + 1)(x − 3). Hence x = 3 or x = −½.
再举一例：解方程 2x² − 5x − 3 = 0。这里 a × c = −6；乘积为 −6、和为 −5 的两个数是 −6 和 1。重写得 2x² − 6x + x − 3 = 2x(x − 3) + 1(x − 3) = (2x + 1)(x − 3)。所以 x = 3 或 x = −½。
8. Common Mistakes | 常见错误
Many students lose marks on factorisation questions because of small but avoidable errors. Watch out for these common traps:
许多学生在因式分解题目上失分，是因为一些细小的、本可避免的错误。请注意以下常见陷阱：
- Forgetting to set the equation equal to zero first. For x² = 3x, do not divide both sides by x; instead write x² − 3x = 0, then x(x − 3) = 0, giving x = 0 or x = 3.
忘记先把方程整理为一边等于零。对于 x² = 3x，不能两边同除以 x；应写成 x² − 3x = 0，再得 x(x − 3) = 0，从而 x = 0 或 x = 3。 - Getting the signs wrong when finding factor pairs. Always check that the sum and product match b and c.
寻找因子对时弄错符号。务必检查”和”与”积”是否分别匹配 b 和 c。 - Losing one solution. A quadratic equation always has two roots when solved by factorisation; check that both brackets are set to zero.
漏掉一个解。用因式分解法求解时，二次方程总有两个根；记得令两个括号都等于零。 - Confusing ‘factorise’ with ‘solve’. If the question says “factorise”, stop at the bracket form; if it says “solve”, give the x values.
混淆”因式分解”与”求解”。题目要求 “factorise” 时，写到括号形式即可；要求 “solve” 时，需给出 x 的值。
9. Exam Tips | 考试技巧
The following tips will help you gain full marks on quadratic factorisation questions:
以下技巧能帮助你在二次因式分解题目中获得满分：
- If the coefficient of x² is negative, factor out −1 first. For example, −x² + 5x − 6 = −(x² − 5x + 6) = −(x − 2)(x − 3).
如果 x² 的系数为负，先提出 −1。例如，−x² + 5x − 6 = −(x² − 5x + 6) = −(x − 2)(x − 3)。 - Recognise perfect squares: x² + 6x + 9 = (x + 3)². This saves time in both factorising and solving.
识别完全平方式：x² + 6x + 9 = (x + 3)²。这能在因式分解和求解时节省时间。 - Always expand your answer mentally to verify it. Substituting the roots back into the original equation is the most reliable check.
随时心算展开来验证。将根代回原方程是最可靠的检验方法。 - If the quadratic does not factorise with integers, do not force it; use the quadratic formula instead. Factorisation is only one method among several.
如果二次式不能用整数因式分解，不要强行分解，改用求根公式。因式分解只是多种方法之一。
10. Practice Questions | 自我练习
Try these questions on your own, then check your answers below.
请先独立完成以下题目，再对照下方答案。
| Question 题目 | Answer 答案 |
| Solve x² + 7x + 12 = 0 | x = −3 or x = −4 |
| Solve x² − 2x = 8 | x = 4 or x = −2 |
| Solve 2x² − x − 3 = 0 | x = 1½ or x =
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