Solving Quadratic Equations by Factorisation | 用因式分解法解二次方程

📚 Solving Quadratic Equations by Factorisation | 用因式分解法解二次方程

Quadratic equations appear in nearly every IGCSE Mathematics examination paper, often in algebra, coordinate geometry and problem-solving questions. This revision guide focuses on solving quadratic equations by factorisation, the most direct method when the expression factorises neatly into linear factors.

二次方程几乎出现在每一份 IGCSE 数学试卷中，常见于代数、坐标几何和应用题。本复习指南聚焦于用因式分解法解二次方程——当表达式能够整齐地分解成一次因式时，这是最直接的方法。

1. What Is a Quadratic Equation? | 什么是二次方程

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the unknown variable is 2. Its general form is:

二次方程是次数为 2 的多项式方程，即未知数的最高次数为 2。它的一般形式为：

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are constants, and x is the variable. The condition a ≠ 0 is essential; if a were zero, the equation would no longer be quadratic but simply linear.

其中 a、b、c 为常数，x 为未知数。条件 a ≠ 0 至关重要；若 a 为零，方程将不再是二次方程，而只是一次方程。

Every quadratic equation has at most two real roots, also called solutions. These roots can be found by factorisation, completing the square, applying the quadratic formula, or reading the x-intercepts from a graph. In this article, we concentrate on factorisation, which is often the quickest and cleanest method when it works.

每一个二次方程至多有两个实根，也称为解。我们可以通过因式分解、配方法、套用求根公式或从图像中读取与 x 轴的交点来求出这些根。本文将集中讲解因式分解法，因为当它适用时，往往是最快捷、最简洁的方法。

2. The Zero Product Property | 零乘积性质

The factorisation method rests on one simple but powerful rule: if the product of two factors is zero, then at least one of the factors must itself be zero.

因式分解法建立在一个简单却非常有力的规则之上：如果两个因数的乘积为零，那么至少有一个因数本身必须为零。

If A × B = 0, then A = 0 or B = 0

This property allows us to split a complicated quadratic equation into two much simpler linear equations. 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.

这一性质使我们能够把一个复杂的二次方程拆分成两个简单得多的一次方程。例如，若 (x − 3)(x + 2) = 0，则要么 x − 3 = 0，要么 x + 2 = 0。解这两个一次方程可得 x = 3 或 x = −2。

It is important to note that this rule works only because the right-hand side is exactly zero. If the product equals any other number, the same conclusion cannot be drawn.

需要注意的是，这条规则仅在右边恰好为零时才成立。如果乘积等于其他任何数，就无法得出同样的结论。

3. Factorising x² + bx + c When a = 1 | 当 a = 1 时因式分解 x² + bx + c

When the coefficient of x² is 1, we look for two numbers whose product equals c and whose sum equals b. This is the most common case in IGCSE papers.

当 x² 的系数为 1 时，我们寻找两个数，使它们的乘积等于 c、和等于 b。这是 IGCSE 试卷中最常见的情形。

The procedure can be summarised in three steps:

整个过程可以概括为三个步骤：

• Step 1: Identify the values of b and c in x² + bx + c.
• 步骤 1：在 x² + bx + c 中确定 b 和 c 的值。
• Step 2: Find two integers m and n such that m × n = c and m + n = b.
• 步骤 2：找到两个整数 m 和 n，使得 m × n = c 且 m + n = b。
• Step 3: Write the factorised form as (x + m)(x + n).
• 步骤 3：将因式分解结果写为 (x + m)(x + n)。

Take x² + 7x + 12 as an example. We need two numbers that multiply to 12 and add to 7. Listing the factor pairs of 12 gives (1, 12), (2, 6) and (3, 4). Only 3 and 4 satisfy both conditions because 3 × 4 = 12 and 3 + 4 = 7.

以 x² + 7x + 12 为例。我们需要两个数相乘得 12、相加得 7。列出 12 的因数对：(1, 12)、(2, 6) 和 (3, 4)。只有 3 和 4 同时满足两个条件，因为 3 × 4 = 12 且 3 + 4 = 7。

Therefore x² + 7x + 12 = (x + 3)(x + 4). Always check your factorisation by expanding the brackets: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12, which matches the original expression.

因此 x² + 7x + 12 = (x + 3)(x + 4)。务必通过展开括号来检验你的因式分解：(x + 3)(

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