Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in almost every exam paper, often in both Paper 2 and Paper 4. In this revision guide, we will explore what a quadratic equation is, how to solve it using different methods, and how to avoid common mistakes.

二次方程是 IGCSE 数学中最重要的话题之一。它几乎出现在每一份考卷中，通常在 Paper 2 和 Paper 4 中都会涉及。在本复习指南中，我们将探讨什么是二次方程、如何用不同方法求解，以及如何避免常见错误。

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 unknown variable is 2. Its general form is ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0.

二次方程是未知数的最高次数为 2 的方程。它的一般形式是 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

For example, 2x² + 3x − 5 = 0 is a quadratic equation, while x³ + x = 0 is not, because the highest power is 3.

例如，2x² + 3x − 5 = 0 是一个二次方程，而 x³ + x = 0 不是，因为最高次数是 3。

Why must a ≠ 0? If a = 0, the equation becomes bx + c = 0, which is a linear equation, not a quadratic one.

为什么必须有 a ≠ 0？如果 a = 0，方程就变成了 bx + c = 0，这是一个一次方程，而不是二次方程。

2. Solving by Factorisation | 因式分解法

Factorisation is the first method you should try, because it is fast and requires no formula. The idea is to write the quadratic expression as a product of two linear factors, then set each factor equal to zero.

因式分解是你应该首先尝试的方法，因为它速度快且不需要套用公式。其思路是将二次表达式写成两个一次因式的乘积，然后令每个因式等于零。

Consider the equation x² + 5x + 6 = 0. We need two numbers that multiply to give 6 and add to give 5. These numbers are 2 and 3. Therefore:

考虑方程 x² + 5x + 6 = 0。我们需要找到两个数，它们相乘等于 6，相加等于 5。这两个数是 2 和 3。因此：

x² + 5x + 6