Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations form one of the most important foundations in IGCSE Mathematics. They appear in algebra, coordinate geometry, word problems and even in harder questions on functions and graphs. Mastering the three core solution methods — factorisation, completing the square and the quadratic formula — is essential for every Cambridge IGCSE candidate.

二次方程是 IGCSE 数学中最基础也最重要的内容之一。它出现在代数、坐标几何、应用题，甚至函数与图像的综合题中。掌握因式分解、配方和求根公式三种核心解法，是每一个剑桥 IGCSE 考生的必修课。

1. What Is a Quadratic Equation? | 什么是一元二次方程

A quadratic equation in one variable can always be written in the standard form ax² + bx + c = 0, where a, b and c are real constants and a ≠ 0. The term ax² is the quadratic term, bx is the linear term and c is the constant term. If a = 0, the equation becomes linear — that is why the condition a ≠ 0 is crucial.

一元二次方程总可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是实常数，且 a ≠ 0。ax² 是二次项，bx 是一次项，c 是常数项。如果 a = 0，方程就退化为一次方程，因此 a ≠ 0 是定义二次方程的关键条件。

The following are all quadratic equations:

以下都是一元二次方程：

• x² − 4x + 3 = 0

• 2x² + 7x − 15 = 0

• x² = 9

Notice that after rearrangement, each equation contains x² as its highest power. In IGCSE exams, you must always rewrite the equation in the form ax² + bx + c = 0 before attempting to solve it.

注意：整理后每个方程的最高次数是 x²。在考试中，必须先把它化成 ax² + bx + c = 0 的形式再求解。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic expression can be written as the product of two linear factors. The key rule is the Zero Product Property: if p × q = 0, then either p = 0 or q = 0. This converts a quadratic equation into two simple linear equations.

当二次表达式能写成两个一次因式的乘积时，因式分解是最快的解法。核心依据是零乘积性质：若 p × q = 0，则 p = 0 或 q = 0。这样就把二次方程化成了两个简单的一次方程。

Example 1: Solve x² − 5x + 6 = 0.

例 1：求解 x² − 5x + 6 = 0。

We look for two numbers that multiply to 6 and add to −5. Since (−2) × (−3) = 6 and (−2) + (−3) = −5, we write:

寻找两个数，乘积为 6，和为 −5。因为 (−2) × (−3) = 6，且 (−2) + (−3) = −5，所以：

(x − 2)(x − 3) = 0

Therefore x − 2 = 0

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