Mastering Quadratic Equations | 精通二次方程

📚 Mastering Quadratic Equations | 精通二次方程

Quadratic equations are a central topic in IGCSE Mathematics. They appear in algebra, graphs, and real-life modelling, and exam questions often combine solving, sketching, and interpreting roots. This guide covers the key methods, common pitfalls, and exam-ready strategies.

二次方程是 IGCSE 数学的核心主题之一。它们出现在代数、函数图像和实际建模中，考试题常常综合考查求解、作图与根的解读。本指南涵盖关键方法、常见错误和备考策略。

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

A quadratic equation is a polynomial equation of degree 2. Its general shape is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

二次方程是一个次数为 2 的多项式方程。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。如果 a = 0，方程就变成一次方程，因此 a ≠ 0 这个条件非常关键。

The highest power of the variable x is 2, which is why we call it quadratic. The word quad means square, and all quadratic equations involve x² as the leading term.

变量 x 的最高次数是 2，因此我们称之为二次。quad 一词表示平方，所有二次方程都含有 x² 作为最高次项。

2. Standard Form and Key Terms | 标准形式与关键术语

Standard form means writing the equation as ax² + bx + c = 0 with the right-hand side equal to zero. For example, 3x² − 5x + 2 = 0 is in standard form, with a = 3, b = −5 and c = 2.

标准形式是指将方程写成 ax² + bx + c = 0，且等号右边为 0。例如，3x² − 5x + 2 = 0 就是标准形式，其中 a = 3，b = −5，c = 2。

The constants a, b and c are called the quadratic coefficient, the linear coefficient and the constant term respectively. Identifying them correctly is vital before using the quadratic formula or the discriminant.

常数 a、b 和 c 分别称为二次项系数、一次项系数和常数项。在使用求根公式或判别式之前，正确识别它们至关重要。

3. Solving by Factorising | 因式分解法求解

Factorising is often the fastest method when the quadratic can be written as a product of two linear brackets. For x² + 5x + 6 = 0, we look for two numbers that multiply to 6 and add to 5: these are 2 and 3.

当二次式可以写成两个一次因式的乘积时，因式分解法通常是最快的方法。对于 x² + 5x + 6 = 0，我们找两个数，使它们相乘得 6、相加得 5：这两个数是 2 和 3。

Therefore, x² + 5x + 6 = (x + 2)(x + 3) = 0. Setting each bracket equal to zero gives x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.

因此，x² + 5x + 6 = (x + 2)(x + 3) = 0。令每个括号分别等于 0，得到 x + 2 = 0 或 x + 3 = 0，所以 x = −2 或 x = −3。

When a is greater than 1, for example 2x² + 7x + 3, we need to find two numbers that multiply to a × c = 6 and add to 7, namely 6 and 1. Then split the middle term and factor by grouping.

当 a 大于 1 时，例如 2x² + 7x + 3，我们需要找两个数，相乘得 a × c = 6，相加得 7，即 6 和 1。然后拆分中间项并用分组分解法。

2x² + 7x + 3 = 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

2x² + 7x + 3 = 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

Finally set each factor to zero: 2x + 1 = 0 gives x = −1/2, and x + 3 = 0 gives x = −3.

最后令每个因式为零：2x + 1 = 0 得 x = −1/2，x + 3 = 0 得 x = −3。

4. The Quadratic Formula | 求根公式法

The quadratic formula solves any quadratic equation in standard form. It is derived by completing the square on ax² + bx + c = 0 and is given by:

求根公式可以求解任何标准形式的二次方程。它是对 ax² + bx + c = 0 进行配方后得到的，公式为：

x = (−b ± √(b² − 4ac)) ÷ (2a)

This formula gives both roots directly. For example, solve 2x² − 4x − 6 = 0. Here a = 2, b = −4 and c = −6. Substitute into the formula:

该公式可以直接给出两个根。例如，求解 2x² − 4x − 6 = 0。这里 a = 2，b = −4，c = −6。代入公式：

x = (4 ±

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