Quadratic Equations: From Factorising to the Quadratic Formula | 二次方程：从因式分解到求根公式

📚 Quadratic Equations: From Factorising to the Quadratic Formula | 二次方程：从因式分解到求根公式

Quadratic equations are a central part of IGCSE Mathematics. They combine algebra, graphs, and problem solving, and exam questions often ask you to choose the most efficient method.

二次方程是 IGCSE 数学的核心内容。它结合了代数、图像和问题求解，考试题常常要求你选择最高效的方法。

This guide covers standard form, factorising, completing the square, the quadratic formula, the discriminant, and common applications.

本指南涵盖标准形式、因式分解法、配方法、求根公式、判别式以及常见应用。

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

A quadratic equation is an equation where the highest power of the unknown is 2. The 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。一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

For example, x² − 3x + 2 = 0 and 2x² + 5x − 3 = 0 are both quadratic equations.

例如，x² − 3x + 2 = 0 和 2x² + 5x − 3 = 0 都是二次方程。

The condition a ≠ 0 is important: if a were zero, the x² term would vanish and the equation would be linear, not quadratic.

条件 a ≠ 0 很重要：如果 a 为零，x² 项就会消失，方程就变成一次方程，而不是二次方程。

2. Standard Form and Identifying Coefficients | 标准形式与识别系数

Before solving, always rearrange the equation into the standard form ax² + bx + c = 0. Collect all terms on one side so the other side equals zero.

在求解之前，一定要先把方程整理成标准形式 ax² + bx + c = 0。把所有项移到同一边，使另一边等于零。

For 3x² − 12x + 9 = 0, the coefficients are a = 3, b = −12 and c = 9.

对于 3x² − 12x + 9 = 0，系数为 a = 3，b = −12，c = 9。

Pay close attention to signs: in x² − 4x − 5 = 0, b is −4 and c is −5, not 4 or 5.

要格外注意符号：在 x² − 4x − 5 = 0 中，b 是 −4，c 是 −5，而不是 4 或 5。

When terms are on both sides, move them carefully. For example, x² = 5x − 6 becomes x² − 5x + 6 = 0, so a = 1, b = −5 and c = 6.

当项在两边时，要小心移项。例如 x² = 5x − 6 变为 x² − 5x + 6 = 0，所以 a = 1，b = −5，c = 6。

3. Solving by Factorising | 因式分解法

If a quadratic factorises easily, this is usually the quickest method. The key idea is the zero-product property: if p × q = 0, then p = 0 or q = 0.

如果二次式容易因式分解，这通常是最快的方法。核心思想是零乘积性质：如果 p × q = 0，那么 p = 0 或 q = 0。

Example: solve x² − 5x + 6 = 0. Factorising gives (x − 2)(x − 3) = 0, so x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

例如：解方程 x² − 5x + 6 = 0。因式分解得 (x − 2)(x − 3) = 0，所以 x − 2 = 0 或 x − 3 = 0，解得 x = 2 或 x = 3。

Always expand the brackets mentally to check the factorisation is correct before writing the final answers.

在写出最终答案前，一定要在心里展开括号，检查因式分解是否正确。

If a factor repeats, such as (x − 3)² = 0, the equation still gives x = 3, but it is a repeated root rather than two different solutions.

如果因子重复，例如 (x − 3)² = 0，方程仍然得到 x = 3，但它是重根，而不是两个不同的解。

4. Factorising When a Is Not 1 | 二次项系数不为 1 时因式分解

When the coefficient of x

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）