📚 188 – Solving Quadratics by Factorising | 188 – 因式分解法解二次方程
Quadratic equations lie at the heart of IGCSE algebra and appear in countless exam questions, from straightforward solving to modelling real-life curves. Factorisation is the quickest and most elegant method when the quadratic can be written as a product of linear factors. This article unpacks the technique step by step, highlights common errors, and provides plenty of practice for Edexcel candidates.
二次方程是IGCSE代数部分的核心，在考试中无处不在，从直接求解到现实曲线建模都会涉及。当二次式能写成两个一次因式的乘积时，因式分解法是最快速、最优雅的解法。本文逐步拆解这一技巧，指出常见错误，并为Edexcel考生提供充足的练习。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which gives the equation its characteristic U-shaped graph called a parabola.
二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次幂为2，这使得它的图像呈U形，称为抛物线。
Examples include x² − 5x + 6 = 0, 2x² + 3x − 2 = 0 and x² − 9 = 0. Notice that each can have up to two real solutions.
例如 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 以及 x² − 9 = 0。注意每个方程最多有两个实数解。
2. Standard Form and Preparation | 标准形式与准备
Before factorising, always rearrange the equation so that one side is zero, written with descending powers of x. For instance, 3x = x² + 2 should become x² − 3x + 2 = 0 once the terms are moved. Expanding any brackets and collecting like terms is an essential first step.
因式分解前，一定要将方程整理成一边为零、并按 x 的降幂排列。比如 3x = x² + 2 移项后应变为 x² − 3x + 2 = 0。展开括号并合并同类项是必不可少的第一步。
ax² + bx + c = 0
Keeping the coefficient a positive often makes factorisation easier, but it is not compulsory.
让系数 a 为正通常会使因式分解更容易，但这不是必须的。
3. The Zero Product Property | 零乘积性质
The whole method rests on a simple rule: if the product of two or more factors is zero, then at least one of the factors must be zero. In symbols, if p × q = 0, then p = 0 or q = 0.
整个方法基于一条简单规则：如果两个或多个因式的乘积为零，那么至少有一个因式为零。用符号表示就是：若 p × q = 0，则 p = 0 或 q = 0。
This powerful idea allows us to break a quadratic into two linear equations once it has been factorised. It works only when one side of the equation is exactly zero, so never skip the rearranging stage.
这个强大的思想让我们在因式分解后将二次方程拆成两个线性方程。它仅在方程的一边恰好为零时才有效，因此千万不要跳过整理方程的步骤。
4. Factorising Simple Quadratics (a = 1) | 分解简单二次三项式（a = 1）
When the coefficient of x² is 1, we look for two numbers that add to give the middle coefficient b and multiply to give the constant term c. The factorised form will be (x + p)(x + q), where p + q = b and pq = c.
当 x² 的系数为1时，我们找两个数，它们的和为中间系数 b，乘积为常数项 c。因式分解后的形式为 (x + p)(x + q)，其中 p + q = b，pq = c。
For example, to factorise x² + 7x + 12, the numbers 3 and 4 satisfy 3 + 4 = 7 and 3 × 4 = 12. Hence,
例如，分解 x² + 7x + 12 时，数字 3 和 4 满足 3 + 4 = 7 和 3 × 4 = 12。因此：
x² + 7x + 12 = (x + 3)(x + 4)
If c is negative, the two numbers will have opposite signs. For x² − 2x − 15, we need two numbers that add to −2 and multiply to −15: these are −5 and +3, giving (x − 5)(x + 3).
如果 c 为负，这两个数异号。对于 x² − 2x − 15，我们需要两个数相加得 −2、相乘得 −15：即 −5 和 +3，得到 (x − 5)(x + 3)。
5. Factorising Quadratics When a ≠ 1 | 当 a ≠ 1 时的因式分解
When the leading coefficient a is not 1, trial and error can be used, but a systematic approach is the ‘ac method’. Multiply a and c, then find two numbers that add to b and multiply to ac. Then split the middle term and factor by grouping.
当首项系数 a 不为 1 时，可以用试错法，但更系统的方法是“ac 法”。先将 a 与 c 相乘，再找两个数，使其和为 b、乘积为 ac。接着拆分中间项并分组分解。
Example: 2x² + 7x + 3. Here a = 2, b = 7, c = 3, so ac = 6. The numbers 6 and 1 give sum 7 and product 6. Rewrite the quadratic as 2x² + 6x + x + 3, then group:
示例：2x² + 7x + 3。这里 a = 2，b = 7，c = 3，因此 ac = 6。数字 6 和 1 满足和为 7、积为 6。将原式改写为 2x² + 6x + x + 3，然后分组：
2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
Another common case is 3x² − 11x − 4. ac = −12, so we need two numbers adding to −11 and multiplying to −12: −12 and 1. Splitting gives 3x² − 12x + x − 4 = 3x(x − 4) + 1(x − 4) = (3x + 1)(x − 4).
另一个常见例子：3x² − 11x − 4。ac = −12，需要两数之和为 −11、积为 −12：−12 和 1。拆分得 3x² − 12x + x − 4 = 3x(x − 4) + 1(x − 4) = (3x + 1)(x − 4)。
6. Special Cases: Difference of Two Squares | 特例：平方差
An expression of the form a² − b² factorises immediately as (a − b)(a + b). This is a vital shortcut in many IGCSE problems.
形如 a² − b² 的式子可直接分解为 (a − b)(a + b)。这是许多IGCSE题目中的重要捷径。
For example, x² − 25 = (x − 5)(x + 5) and 9x² − 16 = (3x − 4)(3x + 4). Always check if a term like 4x² − 81 can be written as (2x)² − 9² before trying more complex methods.
例如，x² − 25 = (x − 5)(x + 5)，9x² − 16 = (3x − 4)(3x + 4)。在尝试更复杂的方法前，务必先检查像 4x² − 81 这样的式子能否写成 (2x)² − 9²。
7. Solving by Factorising – Step-by-Step | 因式分解法求解步骤
Once factorisation is complete, solving is straightforward. Follow these four steps consistently:
一旦完成因式分解，求解就非常直接。始终遵循以下四个步骤：
Step 1: Rearrange the equation to expression = 0.
Step 2: Factorise the quadratic expression fully.
Step 3: Set each factor equal to zero.
Step 4: Solve the resulting simple linear equations.
步骤1：将方程整理为 表达式 = 0。
步骤2：将二次表达式完全因式分解。
步骤3：设每个因式为零。
步骤4：解所得的简单线性方程。
Illustration: Solve x² − 6x + 8 = 0.
Factorise: (x − 2)(x − 4) = 0.
Set each factor to zero: x − 2 = 0 or x − 4 = 0.
Solutions: x = 2 or x = 4.
举例：解 x² − 6x + 8 = 0。
因式分解：(x − 2)(x − 4) = 0。
设每个因式为零：x − 2 = 0 或 x − 4 = 0。
解为：x = 2 或 x = 4。
For 4x² − 25 = 0, recognise the difference of squares: (2x − 5)(2x + 5) = 0 → x = 2.5 or x = −2.5.
对于 4x² − 25 = 0，识别平方差：(2x − 5)(2x + 5) = 0 → x = 2.5 或 x = −2.5。
8. Checking Your Solutions | 检验解
Substituting each solution back into the original equation is a habit that catches sign errors. For x² − 6x + 8 = 0, when x = 2, (2)² − 6(2) + 8 = 4 − 12 + 8 = 0. When x = 4, 16 − 24 + 8 = 0. The check confirms both solutions are correct.
将每个解代回原方程是一种能捕捉符号错误的好习惯。对于 x² − 6x + 8 = 0，当 x = 2 时，(2)² − 6(2) + 8 = 4 − 12 + 8 = 0。当 x = 4 时，16 − 24 + 8 = 0。检验证实两解都正确。
Rapid substitution also reveals whether a factorisation was performed correctly. If a solution does not satisfy the original equation, revisit the factorisation steps.
快速代回还能揭示因式分解是否正确。如果某个解不满足原方程，就需要重新检查分解步骤。
9. Common Mistakes to Avoid | 常见错误
Mistake 1: Forgetting to set the equation to zero first. Factorising x² + 3x = 10 without moving the 10 leads to (x + 5)(x − 2) = 10, which cannot be solved by the zero product rule.
错误1：忘记先将方程设为零。不移动10就对 x² + 3x = 10 因式分解，会得到 (x + 5)(x − 2) = 10，无法用零乘积规则求解。
Mistake 2: Cancelling a variable wrongly. In an equation like x² = 2x, it is tempting to divide both sides by x, losing the solution x = 0. Always bring all terms to one side: x² − 2x = 0 → x(x − 2) = 0 → x = 0 or x = 2.
错误2：错误地约去变量。在 x² = 2x 这样的方程里，人们容易两边同除以 x，从而丢失解 x = 0。务必把所有项移到一边：x² − 2x = 0 → x(x − 2) = 0 → x = 0 或 x = 2。
Mistake 3: Sign errors when splitting the middle term. Double-check the product and sum relationships, especially when a or c are negative.
错误3：拆分中间项时的符号错误。要再三检查积与和的关系，尤其在 a 或 c 为负时。
10. Practice Examples | 练习实例
Work through these typical IGCSE-style equations, factorising and solving them fully.
完成以下典型的IGCSE风格方程，充分进行因式分解并求解。
Example 1: x² − 12x + 35 = 0
实例1：x² − 12x + 35 = 0
Factorisation: (x − 5)(x − 7) = 0 → x = 5 or x = 7.
Example 2: 2x² − 3x − 5 = 0
实例2：2x² − 3x − 5 = 0
ac = −10, numbers −5 and 2. 2x² − 5x + 2x − 5 = x(2x − 5) + 1(2x − 5) = (2x − 5)(x + 1) = 0 → x = 2.5 or x = −1.
Example 3: 9x² − 4 = 0
实例3：9x² − 4 = 0
Difference of squares: (3x − 2)(3x + 2) = 0 → x = 2/3 or x = −2/3.
Example 4: x² + 8x = 0
实例4：x² + 8x = 0
Common factor x: x(x + 8) = 0 → x = 0 or x = −8.
11. Connecting Factorisation to Graphs | 将因式分解与图像联系起来
The solutions of a quadratic equation ax² + bx + c = 0 are the x‑intercepts of the graph y = ax² + bx + c. Factorising gives these roots directly. For y = x² − 4x − 5, the factorised form (x − 5)(x + 1) shows the curve crosses the x‑axis at x = 5 and x = −1. Understanding this link helps when Edexcel questions ask for coordinates where a graph meets the x‑axis.
二次方程 ax² + bx + c = 0 的解就是图像 y = ax² + bx + c 与 x 轴的交点坐标。因式分解直接给出这些根。对于 y = x² − 4x − 5，因式分解为 (x − 5)(x + 1) 表明曲线在 x = 5 和 x = −1 处与 x 轴相交。理解这一联系有助于应对Edexcel考题中询问图像与 x 轴交点坐标的问题。
12. Summary | 总结
To solve quadratics by factorising, rearrange to ax² + bx + c = 0, factorise fully, apply the zero product property, and solve the linear factors. Practise recognising standard patterns such as the difference of two squares and the ac method for non‑monic quadratics. Always check your answers by substitution and never divide by a variable when an equation equals zero. With these techniques, factorisation becomes a reliable and efficient tool for your IGCSE exam.
用因式分解法解二次方程时，先整理成 ax² + bx + c = 0，充分因式分解，运用零乘积性质，再求解线性因式。要练习识别标准模式，如平方差和非首一系的ac方法。始终用代回法检验答案，并绝不在方程等于零时除以变量。掌握这些技巧后，因式分解将成为你在IGCSE考试中可靠且高效的工具。
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