Solving Quadratic Equations by Factorising | 因式分解解二次方程

📚 Solving Quadratic Equations by Factorising | 因式分解解二次方程

In IGCSE Mathematics, quadratic equations appear frequently in algebra papers and are a core skill for higher-tier and extended-tier candidates. A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0. The ability to solve these equations by factorising is one of the most important algebraic techniques. When the quadratic expression can be written as the product of two linear brackets, factorising is often the fastest method. This article explains the factorising method step by step, with worked examples, common errors, and exam tips for IGCSE candidates.

在 IGCSE 数学中，二次方程在代数试卷中频繁出现，是进阶卷和拓展卷考生的核心技能。二次方程是形如 ax² + bx + c = 0 且 a ≠ 0 的方程。通过因式分解求解此类方程是最重要的代数技能之一。当二次式可以写成两个一次括号的乘积时，因式分解通常是最快的方法。本文逐步讲解因式分解法，并配有例题、常见错误以及面向 IGCSE 考生的考试提示。

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 variable is 2. The general form is ax² + bx + c = 0, where a, b and c are constants and a cannot be zero.

二次方程是变量最高次数为 2 的方程。一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a 不能为 0。

ax² + bx + c = 0

If a = 0, the equation becomes linear, not quadratic. In IGCSE questions, you may see quadratics written in different orders, such as x² + 6 = 5x, or with missing terms, such as x² − 4 = 0. Recognising the standard form is the first step in every solution.

若 a = 0，方程就变为一次方程，而不再是二次方程。在 IGCSE 题目中，你可能会看到二次方程以不同顺序书写，例如 x² + 6 = 5x，或者缺项，例如 x² − 4 = 0。识别标准形式是每一个解法的第一步。

2. The Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0

Before factorising, always rewrite the equation in the standard form ax² + bx + c = 0. Move all terms to one side and set the other side equal to zero. For example, x² + 5x = 6 becomes x² + 5x − 6 = 0.

因式分解前，始终先将方程改写为标准形式 ax² + bx + c = 0。把所有项移到一边，令另一边等于零。例如 x² + 5x = 6 应写成 x² + 5x − 6 = 0。

x² + 5x − 6 = 0

If the equation is not in standard form, factorising may lead to the wrong conclusion because the zero product property only applies when one side is zero. In an exam, writing the standard form first also helps you earn method marks.

如果方程不是标准形式，因式分解可能会得出错误结论，因为零乘积性质只在一边为零时适用。在考试中，先写出标准形式也有助于你获得方法分。

3. Why Factorising Works: The Zero Product Property | 因式分解为何有效：零乘积性质

Factorising relies on the zero product property: if two factors multiply to give zero, then at least one of them must be zero. So if (x + p)(x + q) = 0, then either x + p = 0 or x + q = 0.

因式分解依赖零乘积性质：如果两个因式相乘为零，那么其中至少一个因式必须为零。因此，若 (x + p)(x + q) = 0，则要么 x + p = 0，要么 x + q = 0。

(x + p)(x + q) = 0 ⇒ x + p = 0 or x + q = 0

Solving these two linear equations gives the roots of the quadratic. This principle only works when the right-hand side is exactly zero, which is why rearranging into standard form is essential.

解这两个一次方程即可得到二次方程的根。该原理仅在右边恰好为零时成立，因此先整理成标准形式至关重要。

4. Factorising Simple Quadratics with a = 1 | 简单二次式 a = 1 的因式分解

When the coefficient of x² is 1, find two numbers that multiply to give the constant term c and add to give the coefficient b. Write the quadratic as (x + m)(x + n) = 0. The signs must match the signs of b and c.

当 x² 的系数为 1 时，找两个数，使它们的乘积等于常数项 c，和等于系数 b。将二次式写成 (x + m)(x + n) = 0。符号必须与 b 和 c 的符号一致。

x² + bx + c = (x + m)(x + n)

For example, if c is positive and b is positive, both numbers are positive. If c is positive and b is negative, both numbers are negative.

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）