Mastering Quadratic Equations | 二次方程完全攻略

📚 Mastering Quadratic Equations | 二次方程完全攻略

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in many examination questions, either directly or as part of problem solving. This article explains everything you need to know: solving methods, the discriminant, graphs, and exam tips.

二次方程是 IGCSE 数学中最重要的考点之一。无论是直接考查还是作为解题的一部分，二次方程都频繁出现在考试题目中。本文将系统讲解你需要掌握的全部内容：求解方法、判别式、函数图像以及考试技巧。

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

A quadratic equation is a polynomial equation of degree 2. This means the highest power of the unknown variable x is 2. The most general form is ax² + bx + c = 0, where a, b and c are constants and a is not zero.

二次方程是次数为 2 的多项式方程，也就是说未知数 x 的最高次数是 2。它的一般形式是 ax² + bx + c = 0，其中 a、b、c 是常数，且 a 不等于零。

Examples include x² − 3x + 2 = 0, 2x² + x − 1 = 0, and x² = 25. The last one has no linear term, but it is still quadratic because of the x² term.

例如 x² − 3x + 2 = 0、2x² + x − 1 = 0 以及 x² = 25 都是二次方程。最后一个方程没有一次项，但因为有 x² 项，所以仍然是二次方程。

Equations such as x + 3 = 0 or x³ − 1 = 0 are not quadratic: the first is linear and the second is cubic.

像 x + 3 = 0 这样的方程是一次（线性）方程，而 x³ − 1 = 0 是三次方程，它们都不是二次方程。

2. Standard Form and Key Components | 标准形式与关键要素

In the standard form ax² + bx + c = 0, the letter a is called the leading coefficient, b is the coefficient of x, and c is the constant term.

在标准形式 ax² + bx + c = 0 中，a 称为二次项系数，b 是一次项系数，c 是常数项。

The condition a ≠ 0 is essential. If

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