📚 Solving Quadratic Equations by Factorising (Coefficient of x² ≠ 1) | 因式分解法解二次方程（x² 系数不为 1）
In IGCSE Mathematics, quadratic equations appear in many forms, and being able to solve them efficiently is essential. While factorising simple quadratics with a leading coefficient of 1 is straightforward, equations where the coefficient of x² is not 1 require a more structured approach. This article will guide you through the complete process of solving such equations by factorisation, covering the cross-method, grouping, and common exam-style problems. You will also learn how to check your answers and avoid typical mistakes.
在IGCSE数学中，二次方程以多种形式出现，能够高效地求解它们至关重要。虽然首项系数为1的简单二次方程因式分解比较直接，但x²系数不为1的方程则需要一种更有条理的方法。本文将引导你完成用因式分解法解这类方程的全过程，涵盖十字相乘法、分组法以及常见的考试题型。你还将学会如何检验答案并避免典型错误。
1. What Is a Quadratic Equation? | 什么是二次方程？
s, and they represent the x-values where the graph of y = ax² + bx + c crosses the x-axis.
二次方程是次数为2的多项式方程。它的标准形式为 ax² + bx + c = 0，其中 a、b 和 c 为常数，且 a ≠ 0。变量 x 的最高次幂为2。二次方程的解称为根，它们代表图像 y = ax² + bx + c 与 x 轴交点的 x 值。
2. The Factorising Approach | 因式分解法的基本思路
Factorising a quadratic involves writing it as a product of two linear expressions. For example, x² + 5x + 6 factorises to (x + 2)(x + 3). Once factorised, we apply the zero-product property: if the product of two factors is zero, then at least one factor must be zero. This gives us simple linear equations to solve. The same principle holds when a ≠ 1.
因式分解二次方程就是将它写成两个一次表达式的乘积。例如，x² + 5x + 6 可分解为 (x + 2)(x + 3)。分解后即可应用零积性质：若两个因式的乘积为零，则至少有一个因式为零。由此可得出简单的一次方程来求解。当 a ≠ 1 时，原理依然相同。
3. Why the Coefficient of x² Matters | 为什么 x² 的系数很重要
easy because we only need to find two numbers that multiply to c and add to b. However, when a is greater than 1 or negative, the possible combinations increase. We must consider factors of both a and c, and then arrange them so that the cross-products sum to b. This requires a systematic method to avoid guesswork.
当 a = 1 时，因式分解比较简单，只需找到相乘得 c、相加得 b 的两个数即可。但当 a 大于1或为负数时，可能的组合就会增多。我们必须同时考虑 a 和 c 的因数，然后合理排列，使交叉乘积之和等于 b。这就需要采用系统的方法，避免盲目猜测。
4. The Cross-Method (Criss-Cross) | 十字相乘法
The cross-method is a visual tool widely used for factorising quadratics where a ≠ 1. Write the quadratic as ax² + bx + c. Set up two fractions or grids: on the left, write factors of a; on the right, factors of c. Draw cross lines to multiply and add the results. The goal is to find a combination where the sum of the cross-products equals b. This method helps organise trial and error efficiently.
十字相乘法是一种广泛使用的可视化工具，用于分解 a ≠ 1 的二次三项式。将二次式写成 ax² + bx + c。设置两个因式框：左边写 a 的因数，右边写 c 的因数。画交叉线将因数相乘并求和，目标是找到交叉乘积之和等于 b 的组合。这种方法能有效安排试错过程。
| Cross-Method Layout / 十字相乘法结构 | |
|---|---|
| p → q r → s | Factors of a: p, r Factors of c: q, s Cross products: p×s + q×r = b |
5. Step-by-Step Process | 分步流程
tic in standard form ax² + bx + c = 0, ensuring all terms are on one side and equals zero.
2. Identify a, b, and c.
3. List the possible factor pairs of a (both positive and negative).
4. List the possible factor pairs of c (both positive and negative).
5. Try different arrangements in two brackets: (px + q)(rx + s).
6. Expand mentally or using the cross-method to check if the middle term equals bx.
7. Once correct factors are found, set each bracket equal to zero and solve for x.
1. 将二次方程写成标准形式 ax² + bx + c = 0，确保所有项都移到一边且等于零。
2. 确定 a、b 和 c。
3. 列出 a 的所有可能因数对（包括正负数）。
4. 列出 c 的所有可能因数对（包括正负数）。
5. 在 (px + q)(rx + s) 的两个括号中试放不同组合。
6. 通过心算或十字相乘法展开，检查中间项是否等于 bx。
7. 找到正确的因式后，令每个括号等于零，解出 x。
6. Worked Example 1: Positive Coefficients | 例题 1：都是正系数
Solve 2x² + 7x + 3 = 0.
Here a = 2, b = 7, c = 3.
Factors of a: (2,1) or (1,2).
Factors of c: (1,3) or (3,1).
Try (2x + 1)(x + 3): cross-products are 2x×3 = 6x and 1×x = x, sum = 7x. That matches b.
Thus (2x + 1)(x + 3) = 0.
Set 2x + 1 = 0 → x = -½, and x + 3 = 0 → x = -3.
解方程 2x² + 7x + 3 = 0。
此处 a = 2，b = 7，c = 3。
a 的因数组合：(2,1) 或 (1,2)。
c 的因数组合：(1,3) 或 (3,1)。
尝试 (2x + 1)(x + 3)：交叉乘积为 2x×3 = 6x 和 1×x = x，和为 7x，刚好等于 b。
因此 (2x + 1)(x + 3) = 0。
令 2x + 1 = 0 → x = -½，以及 x + 3 = 0 → x = -3。
7. Worked Example 2: Negative Constant Term | 例题 2：常数项为负
Solve 3x² – 5x – 2 = 0.
a = 3, b = -5, c = -2.
Factors of a: (3,1).
Factors of c, considering signs: ( -1, 2 ), ( 1, -2 ), ( -2, 1 ), ( 2, -1 ).
Try (3x + 1)(x – 2): cross-products = 3x×(-2) = -6x, 1×x = x, sum = -5x. Matches b = -5x.
So (3x + 1)(x – 2) = 0.
Solutions: 3x + 1 = 0 → x = -⅓, and x – 2 = 0 → x = 2.
解方程 3x² – 5x – 2 = 0。
a = 3，b = -5，c = -2。
a 的因数：(3,1)。
c 的因数组合考虑正负号：(-1,2), (1,-2), (-2,1), (2,-1)。
尝试 (3x + 1)(x – 2)：交叉乘积为 3x×(-2) = -6x，1×x = x，和为 -5x，符合 b = -5x。
所以 (3x + 1)(x – 2) = 0。
解为：3x + 1 = 0 → x = -⅓，以及 x – 2 = 0 → x = 2。
8. Handling a Negative Leading Coefficient | 处理负的首项系数
If a is negative, it is often easier to factor out –1 first. For example, –2x² + 5x – 3 = 0 becomes –(2x² – 5x + 3) = 0. Then factorise the bracketed quadratic: 2x² – 5x + 3 = (2x – 3)(x – 1). The equation becomes –(2x – 3)(x – 0. Since the negative sign does not affect the zeros, the solutions are x = ³⁄₂ and x = 1.
如果 a 为负数，通常先提取公因数 -1 会更方便。例如 –2x² + 5x – 3 = 0 可化为 –(2x² – 5x + 3) = 0。接着将括号内的二次式因式分解：2x² – 5x + 3 = (2x – 3)(x – 1)。方程变为 –(2x – 3)(x – 1) = 0。因为负号不影响零点的求解，解为 x = ³⁄₂ 和 x = 1。
9. The Factorisation by Grouping Method | 分组分解法
that multiply to ac and add to b. Rewrite bx as the sum of those two numbers’ multiples of x. Then group the first two terms and the last two terms, factor each group, and extract the common binomial factor. This method is especially useful when the numbers are larger.
另一种可靠的方法是分组分解法，适用于所有 a ≠ 1 的二次式 ax² + bx + c。步骤：将 a 与 c 相乘。找到两个数，它们的乘积为 ac，和为 b。将 bx 写成这两个数分别乘以 x 的和。然后对前两项和后两项分组，分别提取公因式，最后提取共同的二项式因式。当数字较大时，这种方法尤其有用。
Example: 6x² + 11x – ac = -60. Two numbers: 15 and -4 (since 15 × -4 = -60, 15 + (-4) = 11). Rewrite: 6x² + 15x – 4x – 10. Group: (6x² + 15x) – (4x + 10) = 3x(2x + 5) – 2(2x + 5) = (3x – 2)(2x + 5).
示例：6x² + 11x – 10。ac = -60。两个数：15 和 -4（因为 15 × -4 = -60，15 + (-4) = 11）。重写为 6x² + 15x – 4x – 10。分组：(6x² + 15x) – (4x + 10) = 3x(2x + 5) – 2(2x + 5) = (3x – 2)(2x + 5)。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Many students forget that the equation must equal zero before factorising. Always rearrange terms. Another common mistake is incorrect sign handling when listing factor pairs of c, especially when c is negative. Always check the middle term sign. Also, do not divide both sides by x unless you are certain x ≠ 0, or you may lose a root. Keep all solutions and verify by substituting back into the original equation.
许多学生忘记在因式分解前必须使方程等于零。务必先移项。另一个常见错误是列出 c 的因数对时符号处理不当，尤其在 c 为负数时。一定要检查中间项的符号。此外，除非确定 x ≠ 0，否则不要两边同除以 x，这样可能会丢掉一个根。保留所有解并代回原方程进行验证。
11. Verifying Your Solutions | 检验你的解
original quadratic equation. For 2x² + 7x + 3 = 0, check x = -½: 2(-½)² + 7(-½) + 3 = 2(¼) – 3.5 + 3 = 0.5 – 0.5 = 0. Check x = -3: 2(9) – 21 + 3 = 18 – quation. Verification not only confirms accuracy but also helps catch sign errors quickly.
将每个根代回原二次方程进行检验。以 2x² + 7x + 3 = 0 为例，检验 x = -½：2(-½)² + 7(-½) + 3 = 2(¼) – 3.5 + 3 = 0.5 – 0.5 = 0。检验 x = -3：2(9) – 21 + 3 = 18 – 18 = 0。两者都满足方程。验证不仅能确认正确性，还能迅速发现符号错误。
12. Summary and Exam Tips | 总结与考试技巧
Factorising ax² + bx + c when a ≠ 1 is a core skill in IGCSE Mathematics. The cross-method and grouping method are both effective; choose whichever you find more reliable. In exams, always show your working, write the factorised form clearly, and present solutions as fractions or decimals as required. With systematic practice, you can solve these quadratics quickly and accurately.
在 a ≠ 1 时分解 ax² + bx + c 是IGCSE数学的核心技能。十字相乘法和分组法都很有效；选择你觉得更可靠的方法。考试中务必要展示解题过程，清晰地写出因式分解形式，并按题目要求将解写成分数或小数。通过系统练习，你将能快速准确地解出这类二次方程。
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