📚 076 Solving Quadratic Equations by Factorisation | 因式分解解二次方程
Mastering quadratic equations is a pivotal skill in IGCSE Mathematics, and factorisation sits at the heart of this topic. Whether you are aiming for a grade 4 or striving for a 9, a solid understanding of how to take a quadratic expression apart and use the zero product property will unlock countless exam questions. This guide walks you through the logic behind factorisation, step-by-step techniques, and common pitfalls, all aligned with the Edexcel IGCSE 076 topic focus.
掌握二次方程是 IGCSE 数学的关键技能，而因式分解处于这一主题的核心。无论你的目标是取得 4 分还是冲击 9 分，扎实理解如何拆解二次表达式并利用零乘积性质，都会帮助你解决大量考题。本指南将带你深入理解因式分解背后的逻辑、分步骤的技巧以及常见误区，全部紧扣 Edexcel IGCSE 076 主题要点。
1. What Is a Quadratic Equation? | 什么是二次方程？
d a ≠ 0. The highest power of the variable x is 2, which gives the equation its characteristic U-shaped graph called a parabola. In IGCSE exams, you will most often see quadratics with integer coefficients, but fractions and decimals also appear.
二次方程是任何能够写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数是 2，因此方程的图像呈特有的 U 形，称为抛物线。在 IGCSE 考试中，最常见的二次方程系数为整数，但分数和小数也会出现。
ions. Recognising the standard form is the first step before any factorisation can be applied.
例如，3x² − 12x + 9 = 0、x² + 5x = 0 以及 2x² − 18 = 0 都是二次方程。在应用任何因式分解方法前，首先要能识别标准形式。
2. Standard Form and Coefficient Rules | 标准形式与系数规则
quadratic into the standard form ax² + bx + c = 0 before attempting to factorise. The term 'a' is called the leading coefficient, 'b' the linear coefficient, and 'c' if a > 1 or negative, extra care is needed. Never forget to set the equation equal to zero – ng without zero on one side will not allow you to use the zero product property correctly.
在尝试因式分解之前，一定要先将二次方程整理成标准形式 ax² + bx + c = 0。a 称为首项系数，b 为一次项系数，c 为常数项。若 a = 1，因式分解通常很简单；若 a > 1 或为负数，就需要格外小心。切勿忘记让方程等于零——如果一边不为零就去因式分解，就无法正确使用零乘积性质。
x² + 4x − 5 = 0, or multiplying by −1: x² − 4x + 5 = 0. Now it is ready for factorisation.
考虑 4x − x² = 5。它看起来不同，但将所有项移到一边得到 −x² + 4x − 5 = 0，或者乘以 −1：x² − 4x + 5 = 0。现在就可以进行因式分解了。
3. The Zero Product Property | 零乘积性质
. This splits a quadratic into two linear equations, which can be solved almost instantly.
利用因式分解解方程的整套方法依赖于一条简单而强大的规则：如果两个表达式的乘积为零，那么至少其中一个表达式必须为零。也就是说，若 (px + q)(rx + s) = 0，则或者 px + q = 0，或者 rx + s = 0。这就把二次方程拆成了两个一元一次方程，几乎立即就能解出。
Without this property, factorising alone would not give solutions. That is why you must always bring all terms to one side and have zero on the other.
没有这条性质，仅靠因式分解无法求得解。这就是为什么必须总是将所有项移到一边，让另一边为零。
4. Factorising Simple Quadratics (a = 1) | 因式分解简单二次式（a=1）
signs: if c is positive, p and q have the same sign as b; if c is negative, p and q have opposite signs and the larger one takes the sign of b.
当 a = 1 时，二次式为 x² + bx + c。进行因式分解时，寻找两个数，使它们的乘积等于 c，和等于 b。若这两个数为 p 和 q，则 x² + bx + c = (x + p)(x + q)。要注意符号：若 c 为正，p 和 q 同号并与 b 同号；若 c 为负，p 和 q 异号，其中绝对值较大的那个数的符号与 b 相同。
2 and add to 7. They are 3 and 4, so (x + 3)(x + 4) = 0 gives solutions x = −3 and x = −4.
例如：x² + 7x + 12。寻找乘积为 12、和为 7 的两个数，得到 3 和 4，因此 (x + 3)(x + 4) = 0，解为 x = −3 和 x = −4。
5. Factorising When a ≠ 1 | 当 a ≠ 1 时的因式分解
For quadratics of the form ax² + bx + c with a > 1, the 'ac method' is a reliable approach. Multiply a and c to get ac. Then find two numbers that multiply to ac and add to b. Rewrite the middle term using these two numbers, split the expression into two pairs, and factorise each pair. Finally, take out the common binomial factor.
对于形如 ax² + bx + c 且 a > 1 的二次式，'ac 方法'是一种可靠的做法。先将 a 与 c 相乘得到 ac。然后找出两个数，其乘积为 ac，其和为 b。用这两个数将中间项改写，把表达式分成两组，分别进行因式分解。最后提取公因式。
− 10. Group: 3x(2x + 5) − 2(2x + 5) = (3x − 2)(2x + 5). Set to zero: x = 2/3 or x = −5/2.
例如：6x² + 11x − 10。这里 a = 6，c = −10，ac = −60。我们需要两个数，其乘积为 −60，和为 11。这两个数是 15 和 −4。改写：6x² + 15x − 4x − 10。分组：3x(2x + 5) − 2(2x + 5) = (3x − 2)(2x + 5)。设为零：x = 2/3 或 x = −5/2。
6. Difference of Two Squares | 平方差公式
d a and c are perfect squares with opposite signs. The expression a² − b² factorises as (a − b)(a + b). Recognising this pattern saves time and avoids unnecessary trial and error. Common IGCSE examples include expressions like x² − 16, 9x² − 25, and even (2x + 1)² − 4.
当 b = 0 且 a 与 c 为异号的完全平方数时，会出现一种特殊模式。a² − b² 可分解为 (a − b)(a + b)。识别出这一模式能节省时间，避免不必要的试错。IGCSE 常见的例子有 x² − 16、9x² − 25，甚至 (2x + 1)² − 4。
n be written as (2x)² − 7² = (2x − 7)(2x + 7) = 0. The solutions are x = 7/2 and x = −7/2.
求解：4x² − 49 = 0。可写成 (2x)² − 7² = (2x − 7)(2x + 7) = 0。解为 x = 7/2 和 x = −7/2。
7. Solving the Equation Step-by-Step | 逐步解方程
SE, both exact answers and simplified fractions are expected; do not leave answers as mixed numbers unless specified.
二次式完成因式分解后，求解就变得轻而易举。写出两个线性方程，然后将 x 分离出来。始终清晰地呈现最终答案，例如 x = 5 或 x = −1。在 IGCSE 考试中，要求给出精确解或简化后的分数；除非题目特别说明，否则不要将答案写成带分数形式。
Example: (x − 2)(3x + 1) = 0. Set x − 2 = 0 ⇒ x = 2; set 3x + 1 = 0 ⇒ 3x = −1 ⇒ x = −1/3. Always write both solutions separated by a comma or the word 'or'.
例如：(x − 2)(3x + 1) = 0。令 x − 2 = 0 ⇒ x = 2；令 3x + 1 = 0 ⇒ 3x = −1 ⇒ x = −1/3。始终将两个解用逗号或“或”字隔开写出。
8. Checking Your Answers | 检验答案
Substituting each solution back into the original quadratic is the best way to verify correctness. Any solution that does not satisfy the original equation indicates an error in factorisation or in solving. This habit not only catches mistakes but also reinforces your understanding of the relationship between roots and factors.
将每个解代回原二次方程是验证正确性的最佳方法。任何不满足原方程的解都意味着因式分解或求解过程有误。这一习惯不仅能帮助发现错误，也能加深你对方程根与因式之间关系的理解。
= 0. Valid. For the other root x = −1.5: 2(2.25) + 1.5 − 6 = 4.5 + 1.5 − 6 = 0. Both work.
对于 2x² − x − 6 = 0 中 x = 2：2(2)² − 2 − 6 = 8 − 2 − 6 = 0，成立。另一个根 x = −1.5：2(2.25) + 1.5 − 6 = 4.5 + 1.5 − 6 = 0，同样成立。
9. Common Mistakes to Avoid | 常见错误要避免
Many students lose marks by overlooking simple errors. The most frequent mistakes include: forgetting to set the equation to zero first; incorrectly computing the product ac; mixing up the signs of the numbers; factorising but forgetting to solve the two resulting equations; and not simplifying fractions. Always double-check the signs in the factors – (x − 5)(x + 3) expands to x² − 2x − 15, not x² − 8x − 15.
许多学生因忽视简单的错误而失分。最常见的错误包括：忘记先将方程设为零；ac 乘积计算错误；混淆数字的正负号；完成了因式分解却忘记解随后得到的两个方程；以及未简化分数。务必仔细检查因式中的符号——(x − 5)(x + 3) 展开后是 x² − 2x − 15，而不是 x² − 8x − 15。
4x, dividing by x gives x = 4, but x = 0 is also a solution. Always factorise: x² − 4x = 0 ⇒ x(x − 4) = 0, giving x = 0 or x = 4.
另一个陷阱：两边同除以 x。若 x² = 4x，两边除以 x 得到 x = 4，但 x = 0 也是一个解。务必使用因式分解：x² − 4x = 0 ⇒ x(x − 4) = 0，得到 x = 0 或 x = 4。
10. Worked Examples (Step by Step) | 分步实例详解
Let's work through three typical IGCSE questions.
我们来仔细拆解三个典型的 IGCSE 题目。
Example 1: 0. Find two numbers with product −12 and sum −1: they are −4 and 3. So (x − 4)(x + 3) = 0 ⇒ x = 4 or x = −3.
例1：解 x² − x − 12 = 0。寻找乘积为 −12、和为 −1 的两个数：得到 −4 与 3。因此 (x − 4)(x + 3) = 0 ⇒ x = 4 或 x = −3。
Example 2: + 15x − 2x − 6 = 5x(x + 3) − 2(x + 3) = (5x − 2)(x + 3) = 0. Solutions: x = 2/5 or x = −3.
例2：解 5x² + 13x − 6 = 0。ac = 5 × (−6) = −30。乘积为 −30、和为 13 的两个数：15 与 −2。改写：5x² + 15x − 2x − 6 = 5x(x + 3) − 2(x + 3) = (5x − 2)(x + 3) = 0。解为 x = 2/5 或 x = −3。
Example 3: Solve 16x² − 25 = 0. Difference of squares: (4x)² − 5² = (4x − 5)(4x + 5) = 0 ⇒ x = 5/4 or x = −5/4.
例3：解 16x² − 25 = 0。平方差：(4x)² − 5² = (4x − 5)(4x + 5) = 0 ⇒ x = 5/4 或 x = −5/4。
11. When Factorisation Fails – The Quadratic Formula | 因式分解失效时 – 二次公式
Not all quadratics factorise neatly. When the discriminant b² − 4ac is not a perfect square, or when you simply cannot spot the factors, use the quadratic formula:
并非所有二次式都能整齐地因式分解。当判别式 b² − 4ac 不是一个完全平方数，或者你就是无法找到因数时，可使用二次公式：
x = [ −b ± √b² − 4ac ] / (2a)
This formula is the safety net that guarantees you can solve any quadratic. However, factorisation remains the faster method when it works and is heavily tested in the 076 topic area. Always try factorisation first; if stuck after a reasonable attempt, switch to the formula.
这个公式是确保你能解出任何二次方程的安全网。不过，在可行的情况下，因式分解仍然是更快的方法，也是 076 主题重点考查的内容。始终优先尝试因式分解；如果经过合理尝试后仍受阻，再改用公式。
12. Real-World Connections and Exam Tips | 实际应用与应试技巧
Quadratic equations model a wide range of real-life situations: projectile motion, area problems, economics profit functions, and more. In IGCSE, word problems often require you to form a quadratic from a given scenario and then solve it by factorisation. Always define your variable, translate the words into an equation, and check that the solutions make sense in context (negative lengths are impossible, for example).
二次方程可用来建立众多现实情境的模型：抛体运动、面积问题、经济学利润函数等等。在 IGCSE 中，应用题往往要求你根据给定的情境列出二次方程，然后用因式分解求解。务必先定义变量，将文字翻译成方程，并检验解在实际背景下是否有意义（例如长度不可能为负值）。
- Top exam tip: Show full factorisation steps; examiners award method marks even if the final answer is wrong.
- 最佳应试技巧：写出完整的因式分解步骤；即使最终答案错误，考官也会评给过程分。
- If a question asks for 'exact solutions', leave answers in surd or fraction form, not rounded decimals.
- 如果题目要求“精确解”，请将答案保留为根号或分数形式，不要取小数点后近似值。
- Sketch a quick graph to check if the roots match the intercepts on the x-axis – this visual check catches sign errors.
- 快速画一个草图，检查方程的根是否与 x 轴截距吻合——这一视觉检查能帮你发现符号错误。
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