Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. Whether you are taking Cambridge IGCSE or Edexcel, you will be asked to solve equations of the form ax² + bx + c = 0 using at least one of the four methods covered in this article. Mastering these techniques saves time in your exam and builds the algebra skills needed for later topics such as functions, graphs, and calculus.

一元二次方程是 IGCSE 数学中考查频率最高的知识点之一。无论你参加的是剑桥 IGCSE 还是 Edexcel 考试，都会遇到 ax² + bx + c = 0 形式的方程。掌握本文介绍的四种解法，不仅能帮你节省考试时间，还能为函数、图像和微积分等后续内容打下扎实的代数基础。

1. What Is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is a polynomial equation of degree 2. Its general form is:

一元二次方程是指最高次数为 2 的多项式方程，它的一般形式为：

ax² + bx + c = 0, where a ≠ 0

Here x is the unknown variable, and a, b, c are constants. The highest power of x is 2, which is why it is called ‘quadratic’. If a = 0, the equation becomes linear (bx + c = 0), so the condition a ≠ 0 is essential.

其中 x 是未知数，a、b、c 是常数。因为 x 的最高次数是 2，所以称为“二次”。若 a = 0，方程就退化为一次方程 bx + c = 0，因此必须强调 a ≠ 0。

Examples include: x² − 3x + 2 = 0, 2x² + 5x − 1 = 0, x² = 16.

例如：x² − 3x + 2 = 0、2x² + 5x − 1 = 0、x² = 16。

Note that the last example can be rewritten as x² − 16 = 0, which fits the standard form. Before solving any quadratic, it is often useful to rearrange it into the form ax² + bx + c = 0.

注意最后一个例子可改写为 x² − 16 = 0，符合标准形式。解任何二次方程之前，通常应先将方程整理成 ax² + bx + c = 0 的形式。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the equation has simple integer solutions. The idea is to write ax² + bx + c as a product of two brackets, then use the fact that if AB = 0, then either A = 0 or B = 0.

当方程有简单的整数解时，因式分解是最快的方法。其核心思想是把 ax² + bx + c 写成两个括号相乘的形式，再利用“若 AB = 0，则 A = 0 或 B = 0”这一性质。

Example 1: Solve x² − 5x + 6 = 0.

例 1：解方程 x² − 5x + 6 = 0。

Find two numbers that multiply to 6 and add to −5. These numbers are −2 and −3, so:

找两个数，使它们的乘积为 6 且和为 −5。这两个数是 −2 和 −3，于是：

x² − 5x + 6 = (x − 2)(x − 3) = 0

Therefore x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

因此 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

Example 2: Solve 2x² + 5x − 3 = 0.

例 2：解方程 2x² + 5x − 3 = 0。

For a quadratic with a leading coefficient not equal to 1, we look for two numbers whose product is a × c = 2 × (−3) = −6 and whose sum is b = 5. These are 6 and −1, so we split the middle term:

当二次项系数不为 1 时，先找两个数，使它们的乘积等于 a × c = 2 × (−3) = −6，且它们的和等于 b = 5。这两个数是 6 和 −1，于是分裂中间项：

2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (2x − 1)(x + 3)

So (2x − 1)(x + 3) = 0, giving x = ½ or x = −3.

所以 (2x − 1)(x + 3) = 0，解得 x = ½ 或 x = −3。

Also remember the difference of two squares: x² − 9 = (x − 3)(x + 3) = 0, so x = ±3. This form appears frequently in exams.

还要记住平方差公式：x² − 9 = (x − 3)(x + 3) = 0，所以 x = ±3。这种形式在考试中经常出现。

3. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0, the solutions are:

当因式分解困难或无法分解时，使用求根公式。对于 ax² + bx + c = 0，解为：

x = (−b ± √(b² − 4ac)) / (2a)

This formula works for every quadratic equation. It is printed in the Edexcel formula book but must be memorised for Cambridge IGCSE. You should substitute a, b and c carefully, paying close attention to negative signs.

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