📚 Quadratic Equations: Solving by Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式解法
Quadratic equations are central to IGCSE Mathematics. This article explains how to solve them by factorising, completing the square, and using the quadratic formula. You will also learn how the discriminant reveals the nature of the roots and how to choose the best method for each question.
二次方程是 IGCSE 数学的核心内容。本文将讲解如何通过因式分解、配方法和求根公式求解二次方程。你还会学习判别式如何揭示根的性质，以及如何为每道题选择最佳方法。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation of degree 2, meaning the highest power of the unknown variable x is 2. Quadratic equations appear in problems involving area, projectile motion, and many optimisation tasks.
二次方程是次数为 2 的方程，即未知数 x 的最高次数为 2。二次方程出现在涉及面积、抛体运动和许多优化任务的问题中。
General form: ax² + bx + c = 0, a ≠ 0
The constants a, b and c are called coefficients. The condition a ≠ 0 is essential because if a = 0, the equation becomes linear, not quadratic. For example, x² − 5x + 6 = 0 and 2x² + 3x − 2 = 0 are quadratic equations.
常数 a、b、c 称为系数。条件 a ≠ 0 非常重要，因为如果 a = 0，方程就变成一次方程，而不是二次方程。例如 x² − 5x + 6 = 0 和 2x² + 3x − 2 = 0 都是二次方程。
- x² − 5x + 6 = 0 x² 减 5x 加 6 等于 0
- 2x² + 3x − 2 = 0 2x² 加 3x 减 2 等于 0
- x² − 9 = 0 x² 减 9 等于 0
When the quadratic expression is plotted as y = ax² + bx + c, the solutions of the equation are the x-values where the graph crosses the x-axis. These values are also called roots or x-intercepts.
当二次式绘制为 y = ax² + bx + c 时，方程的解就是图像与 x 轴相交处的 x 值。这些值也称为根或 x 轴截距。
2. Standard Form and Key Terms | 标准形式与关键术语
Before solving any quadratic equation, rearrange all terms onto one side so that the other side is exactly zero. The standard form makes it easy to identify the coefficients a, b and c correctly.
在求解任何二次方程之前，先把所有项移到一边，使另一边恰好为零。标准形式有助于正确识别系数 a、b、c。
ax² + bx + c = 0
| Term | Meaning | 中文 |
| a | leading coefficient | 二次项系数 |
| b | linear coefficient | 一次项系数 |
| c | constant term | 常数项 |
| roots | solutions of the equation | 方程的根 |
| x-intercepts | where the graph crosses the x-axis | 图像与 x 轴交点 |
In the context of y = ax² + bx + c, the roots, solutions and x-intercepts all refer to the same values obtained by setting y = 0. Understanding this link between algebra and graph is essential for IGCSE paper questions.
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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