Mastering Quadratic Equations: Factorising, Completing the Square, and the Quadratic Formula | 掌握二次方程：因式分解、配方法与求根公式

📚 Mastering Quadratic Equations: Factorising, Completing the Square, and the Quadratic Formula | 掌握二次方程：因式分解、配方法与求根公式

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to problem solving in geometry and physics. Being able to solve them confidently using factorising, completing the square, and the quadratic formula is a core skill that examiners test every year. This article breaks down the methods step by step, highlights common mistakes, and provides revision-style examples.

二次方程贯穿 IGCSE 数学大纲，从代数、图像到几何与物理问题求解都有出现。能够熟练运用因式分解、配方法和求根公式解二次方程是每年考官都会考查的核心技能。本文逐步拆解这些方法，指出常见错误，并提供复习型例题。

1. Recognising a Quadratic Equation | 认识二次方程

A quadratic equation in one variable is any equation that can be rearranged into the form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. The condition a ≠ 0 is essential; if a = 0, the equation becomes linear.

一元二次方程是任何可以整理为 ax² + bx + c = 0 形式的方程，其中 a、b、c 为实数且 a ≠ 0。条件 a ≠ 0 至关重要；如果 a = 0，方程就变成一次方程。

For example, x² − 4x + 3 = 0 is quadratic with a = 1, b = −4 and c = 3. However, x² = 5x − 6 is also quadratic because it can be rearranged to x² − 5x + 6 = 0.

例如，x² − 4x + 3 = 0 是二次方程，a = 1、b = −4、c = 3。而 x² = 5x − 6 也是二次方程，因为它可以整理为 x² − 5x + 6 = 0。

In IGCSE questions, quadratic equations may be hidden inside word problems, fractions, or formulae. Always bring them to standard form before choosing a solution method.

在 IGCSE 题目中，二次方程可能隐藏在文字题、分式或公式中。务必先将其化为标准形式，再选择解法。

2. Solving by Factorising | 因式分解法

The factorising method works when the quadratic expression ax² + bx + c can be written as a product of two linear factors. For a simple quadratic with a = 1, we need two numbers whose product is c and whose sum is b.

当二次式 ax² + bx + c 可以写成两个一次因式的乘积时，因式分解法非常有效。对于 a = 1 的简单二次式，我们需要找到两个数，它们的乘积为 c，和为 b。

Example: solve x² + 5x + 6 = 0. We need two numbers with product 6 and sum 5; they are 2 and 3. Thus x² + 5x + 6 = (x + 2)(x + 3) = 0, giving x = −2 or x = −3.

例题：解 x² + 5x + 6 = 0。我们需要两个数，乘积为 6，和为 5；它们是 2 和 3。因此 x² + 5x + 6 = (x + 2)(x + 3) = 0，得到 x = −2 或 x = −3。

When a ≠ 1, use the ‘ac method’: find two numbers with product ac and sum b, split the middle term, then factor by grouping.

当 a ≠ 1 时，可使用 ac 法：找出乘积为 ac、和为 b 的两个数，拆开中间项，然后分组分解。

Example: 2x² + 7x + 3 has ac = 6 and b = 7. The numbers are 6 and 1, so 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

例题：2x² + 7x + 3 中 ac = 6，b = 7。所需两个数为 6 和 1，因此 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

3. The Zero Product Property | 零乘积性质

The reason factorising works is the zero product property: if the product of

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