Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, graphs, geometry, and real-life problems. On the Edexcel IGCSE exam, questions on quadratics can be worth several marks and often appear in both Paper 1 and Paper 2. This article gives you a complete guide to solving and understanding quadratic equations, following the Edexcel IGCSE specification.

二次方程是IGCSE数学大纲中最重要的内容之一。它们出现在代数、图像、几何以及现实问题中。在Edexcel IGCSE考试中，二次方程相关题目往往分值较高，并常出现在Paper 1和Paper 2中。本文依据Edexcel IGCSE考纲，为你全面讲解如何求解和理解二次方程。

You will learn four main methods of solving quadratic equations: factorising, completing the square, and using the quadratic formula. You will also learn about the discriminant, how to sketch graphs, and how to apply these skills to word problems.

你将学习解二次方程的四种主要方法：因式分解、配方法和求根公式。你还会学习判别式、如何画图以及如何将这些技能应用到应用题中。

1. Introduction to Quadratic Equations | 二次方程简介

A quadratic equation is a polynomial equation of degree 2. Its general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which means the graph of a quadratic function is a curve called a parabola.

二次方程是最高次数为2的多项式方程。其一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数是2，这意味着二次函数的图像是一条称为抛物线的曲线。

The word ‘quadratic’ comes from the Latin word ‘quadratus’, meaning square. This is because the variable x is squared. In the Edexcel IGCSE syllabus, quadratic equations are essential for solving problems involving areas, projectile motion, and optimisation.

“quadratic”一词源自拉丁语“quadratus”，意为“平方”。这是因为变量 x 被平方了。在Edexcel IGCSE大纲中，二次方程对于解决面积、抛体运动和优化问题必不可少。

2. Standard Form of a Quadratic | 二次方程的标准形式

Every quadratic equation can be written in the standard form:

每个二次方程都可以写成标准形式：

ax² + bx + c = 0

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. For example, 3x² − 5x + 2 = 0 has a = 3, b = −5 and c = 2. If a = 0, the equation becomes linear, not quadratic, because the x² term disappears.

这里，a 是 x² 的系数，b 是 x 的系数，c 是常数项。例如，方程 3x² − 5x + 2 = 0 中，a = 3，b = −5，c = 2。如果 a = 0，方程就变成一次方程，而不是二次方程，因为 x² 项消失了。

Always rearrange the equation into this form before solving. Do not try to factor

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