📚 Solving Quadratic Equations by Factorisation | 因式分解法解二次方程
Quadratic equations appear frequently in both pure and applied mathematics. In the Edexcel IGCSE syllabus, solving a quadratic equation by factorisation is a core skill that tests your ability to recognise patterns, manipulate algebraic expressions, and apply the zero-product property. This article will guide you through the essential steps, common pitfalls, and exam-style examples.
二次方程在纯数学和应用数学中都频繁出现。在 Edexcel IGCSE 大纲中，用因式分解法解二次方程是一项核心技能，它考验你对模式的识别、代数式的变形以及零乘积性质的运用。本文将引导你掌握关键步骤、常见陷阱以及考试风格的例题。
1. The General Form of a Quadratic Equation | 二次方程的一般形式
A quadratic equation in one variable is an equation that can be written in the form ax² + bx + c = 0, where a, b and c are real numbers, and a ≠ 0. The highest power of the variable is 2, which is why it is called ‘quadratic’.
一元二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是实数，且 a ≠ 0。变量的最高次数是 2，因此称为“二次”。
For example, x² − 5x + 6 = 0 is a quadratic equation because it fits the general form with a = 1, b = −5 and c = 6.
例如，x² − 5x + 6 = 0 是一个二次方程，因为它符合一般形式，其中 a = 1，b = −5，c = 6。
General form: ax² + bx + c = 0 (a ≠ 0)
2. The Zero-Product Property | 零乘积性质
The zero-product property states that if the product of two expressions is zero, then at least one of the expressions must be zero. In symbols, if p × q = 0, then p = 0 or q = 0. This property is the mathematical foundation that allows us to solve factorised quadratic equations.
零乘积性质指出：如果两个表达式的乘积为零，那么至少有一个表达式必须为零。用符号表示，若 p × q = 0，则 p = 0 或 q = 0。这个性质是我们能够解因式分解后二次方程的数学基础。
For example, if (x − 3)(x + 2) = 0, then either x − 3 = 0 or x + 2 = 0. Solving these two linear equations gives x = 3 or x = −2.
例如，若 (x − 3)(x + 2) = 0，那么要么 x − 3 = 0，要么 x + 2 = 0。解这两个一次方程得到 x = 3 或 x = −2。
If (x − a)(x − b) = 0, then x = a or x = b.
3. Factorising Quadratic Expressions (a = 1) | 因式分解二次表达式（a = 1）
When the coefficient of x² is 1, we look for two numbers whose sum is b and whose product is c. To factorise x² + bx + c, find integers m and n such that m + n = b and mn = c. Then x² + bx + c = (x + m)(x + n).
当 x² 的系数为 1 时，我们寻找两个数，使它们的和为 b，积为 c。要将 x² + bx + c 因式分解，找到整数 m 和 n，满足 m + n = b 且 mn = c。于是 x² + bx + c = (x + m)(x + n)。
Example: Factorise x² − 7x + 12. We need numbers that multiply to 12 and add to −7. The numbers are −3 and −4. Therefore x² − 7x + 12 = (x − 3)(x − 4).
例如：因式分解 x² − 7x + 12。我们需要乘积为 12、和为 −7 的两个数。这两个数是 −3 和 −4。因此 x² − 7x + 12 = (x − 3)(x − 4)。
4. Factorising Quadratic Expressions (a ≠ 1) | 因式分解二次表达式（a ≠ 1）
When the coefficient of x² is not 1, we use a method such as ‘factor by grouping’ or ‘trial and error’. For ax² + bx + c, multiply a and c to find the product ac, then find two numbers that multiply to ac and add to b. Rewrite the middle term using these two numbers, then factor by grouping.
当 x² 的系数不为 1 时，我们使用“分组分解法”或“尝试法”。对于 ax² + bx + c，先将 a 和 c 相乘得到 ac，然后找两个数相乘等于 ac、相加等于 b。用这两个数重写中间项，然后进行分组因式分解。
Example: Factorise 2x² + 7x + 3. Here a = 2, b = 7, c = 3, so ac = 6. Two numbers with product 6 and sum 7 are 1 and 6. Rewrite: 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3).
例如：因式分解 2x² + 7x + 3。这里 a = 2，b = 7，c = 3，所以 ac = 6。乘积为 6、和为 7 的两个数是 1 和 6。重写：2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)。
5. Solving Equations by Setting Each Factor to Zero | 令每个因式为零求解方程
Once the quadratic expression is fully factorised, we apply the zero-product property. Write the equation in the form (px + q)(rx + s) = 0. Then set each factor equal to zero and solve the resulting linear equation.
当二次表达式被完全因式分解后，我们应用零乘积性质。将方程写成 (px + q)(rx + s) = 0 的形式。然后令每个因式等于零，并解所得的一次方程。
Example: Solve 2x² + 7x + 3 = 0. Factorising gives (2x + 1)(x + 3) = 0. Hence 2x + 1 = 0 or x + 3 = 0. Solving gives x = −½ or x = −3.
例如：解 2x² + 7x + 3 = 0。因式分解得 (2x + 1)(x + 3) = 0。因此 2x + 1 = 0 或 x + 3 = 0。解得 x = −½ 或 x = −3。
Always check your solutions by substituting them back into the original equation.
6. Difference of Two Squares | 平方差公式
A special case of factorisation is the difference of two squares: a² − b² = (a − b)(a + b). This can be used to solve quadratic equations of the form x² − k = 0, where k is a perfect square, or more general forms such as 9x² − 16 = 0.
因式分解的一个特例是平方差公式：a² − b² = (a − b)(a + b)。它可用于解形式为 x² − k = 0（其中 k 是完全平方数）的二次方程，或更一般的形式，如 9x² − 16 = 0。
Example: Solve x² − 25 = 0. Since x² − 25 = (x − 5)(x + 5) = 0, we get x = 5 or x = −5.
例如：解 x² − 25 = 0。因为 x² − 25 = (x − 5)(x + 5) = 0，所以得到 x = 5 或 x = −5。
This method is faster than expanding and works whenever the quadratic has no x term and both terms are perfect squares.
这种方法比展开更快，并且只要二次方程没有 x 项且两项都是完全平方数即可使用。
7. Perfect Square Trinomials | 完全平方三项式
A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. Recognising this form allows immediate factorisation. The two standard forms are a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)².
完全平方三项式是可以写成二项式平方的二次表达式。识别这种形式可以立即进行因式分解。两种标准形式是 a² + 2ab + b² = (a + b)² 和 a² − 2ab + b² = (a − b)²。
Example: Solve x² + 6x + 9 = 0. This can be factored as (x + 3)² = 0, so the only solution is x = −3. This is called a repeated root.
例如：解 x² + 6x + 9 = 0。这可以分解为 (x + 3)² = 0，因此唯一的解是 x = −3。这称为重根。
Similarly, x² − 10x + 25 = (x − 5)² = 0 gives x = 5 only.
类似地，x² − 10x + 25 = (x − 5)² = 0 只给出 x = 5。
8. Rearranging Equations into Standard Form | 将方程化为标准形式
Sometimes the quadratic equation is not initially in the form ax² + bx + c = 0. It may contain brackets, fractions, or have terms on both sides. Before factorising, expand and simplify, then move all terms to one side so that the other side is zero.
有时二次方程最初并不具有 ax² + bx + c = 0 的形式。它可能包含括号、分数，或两边都有项。在因式分解之前，先展开并化简，然后将所有项移到一侧，使另一侧为零。
Example: Solve x² = 5x. Do not divide by x! Instead, rewrite as x² − 5x = 0, then factorise x(x − 5) = 0, giving x = 0 or x = 5.
例如：解 x² = 5x。不要除以 x！应重写为 x² − 5x = 0，然后因式分解 x(x − 5) = 0，得到 x = 0 或 x = 5。
Example: Solve (x − 2)(x + 3) = 14. Expand: x² + x − 6 = 14. Then x² + x − 20 = 0, factorise to (x + 5)(x − 4) = 0, so x = −5 or x = 4.
例如：解 (x − 2)(x + 3) = 14。展开：x² + x − 6 = 14。然后 x² + x − 20 = 0，因式分解为 (x + 5)(x − 4) = 0，所以 x = −5 或 x = 4。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One common mistake is forgetting to set the equation to zero before factorising. Another is dividing both sides by a variable, which loses solutions. Also, make sure to check signs carefully when finding factor pairs.
一个常见错误是忘记在因式分解前将方程化为零。另一个错误是两边同时除以一个变量，这会丢失解。此外，在寻找因子对时务必小心符号。
- Always bring all terms to one side first. | 始终先将所有项移到一侧。
- Never divide by a variable expression. | 切勿除以含有变量的表达式。
- Check mental arithmetic: the product of the constant terms must equal c. | 检查心算：常数项的乘积必须等于 c。
- Substitute your answers back to verify. | 将答案代回原方程进行验证。
For example, in factorising x² − 5x + 6, a common error is writing (x − 1)(x − 6) because the sum would be −7, not −5. The correct factor pair is (−2, −3).
例如，在因式分解 x² − 5x + 6 时，常见错误是写成 (x − 1)(x − 6)，因为其和为 −7 而不是 −5。正确的因子对是 (−2, −3)。
10. Exam-Style Example with Full Solution | 考试风格例题与完整解答
Let us work through a typical Edexcel IGCSE question step by step.
让我们一步步完成一道典型的 Edexcel IGCSE 题目。
Solve 3x² − 8x − 3 = 0.
Step 1: Identify a = 3, b = −8, c = −3. Multiply a and c: 3 × (−3) = −9.
Step 2: Find two numbers whose product is −9 and whose sum is −8. These are −9 and 1.
Step 3: Rewrite the middle term: 3x² − 9x + x − 3.
Step 4: Factor by grouping: 3x(x − 3) + 1(x − 3) = (3x + 1)(x − 3).
Step 5: Set each factor to zero: 3x + 1 = 0 ⇒ x = −⅓; x − 3 = 0 ⇒ x = 3.
步骤 1：确定 a = 3，b = −8，c = −3。将 a 与 c 相乘：3 × (−3) = −9。
步骤 2：找两个数，乘积为 −9，和为 −8。这两个数是 −9 和 1。
步骤 3：重写中间项：3x² − 9x + x − 3。
步骤 4：分组因式分解：3x(x − 3) + 1(x − 3) = (3x + 1)(x − 3)。
步骤 5：令每个因式为零：3x + 1 = 0 ⇒ x = −⅓；x − 3 = 0 ⇒ x = 3。
Both values satisfy the original equation, so the solution set is x = −⅓ and x = 3.
两个值均满足原方程，因此解集为 x = −⅓ 和 x = 3。
11. Summary of the Method | 方法总结
To solve a quadratic equation by factorisation, follow this sequence: expand and rearrange to standard form, factorise the quadratic expression, apply the zero-product property, solve the resulting linear equations, and finally check your answers.
要用因式分解法解二次方程，按以下顺序进行：展开并整理为标准形式，对二次表达式进行因式分解，应用零乘积性质，解所得的一次方程，最后检查答案。
Standard form → Factorise → Zero-product → Solve → Check
With regular practice, factorisation becomes a quick and reliable method for many quadratic equations, especially when the roots are rational numbers.
通过经常练习，因式分解将成为解许多二次方程（尤其是根为有理数时）快捷且可靠的方法。
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