Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are among the most frequently tested topics in IGCSE Mathematics (Syllabus 0580). Whether you are taking the Core or Extended paper, you will encounter quadratic equations in algebra, coordinate geometry, and problem-solving questions. This guide explores every method you need: factorisation, the quadratic formula, completing the square, and graphical interpretation.

一元二次方程是 IGCSE 数学（教学大纲 0580）中考查最频繁的内容之一。无论你参加 Core 还是 Extended 考试，都会在代数、坐标几何和解答题中遇到一元二次方程。本指南将深入探讨你需要的所有方法：因式分解法、求根公式、配方法和图像法。

1. The General Form | 一般形式

A quadratic equation is a polynomial equation of degree 2. The general form is:

一元二次方程是次数为 2 的多项式方程，其一般形式为：

ax² + bx + c = 0 （其中 a ≠ 0）

Here, a, b and c are constants. The coefficient a of x² must be non-zero; if a = 0, the equation becomes linear. The coefficients b and c may be zero, giving forms such as ax² = 0, ax² + c = 0, or ax² + bx = 0. Every quadratic equation has at most two solutions, called roots, which may be real and distinct, real and equal, or non-real.

其中 a、b、c 为常数。x² 的系数 a 不能为零；若 a = 0，方程退化为一次方程。b 和 c 可以为 0，从而得到 ax² = 0、ax² + c = 0 或 ax² + bx = 0 等形式。每个一元二次方程最多有两个解（称为根），它们可能是两个相异实根、一个二重实根，或无实根。

2. Solving by Factorisation | 因式分解法

When the quadratic expression can be written as the product of two linear brackets, factorisation is the fastest method. The procedure is as follows:

当二次式可以写成两个一次因式相乘时，因式分解法是最快捷的方法。步骤如下：

• Write the equation in the standard form ax² + bx + c = 0 — 将方程化为标准形式 ax² + bx + c = 0

• Factorise the left-hand side into two brackets — 将左边分解为两个括号相乘

• Apply the Zero Product Property: set each bracket to zero — 利用零乘积性质：令每个括号等于零

• Solve the two resulting linear equations — 解两个得到的一次方程

Example / 例： Solve / 解 x² − 5x + 6 = 0

We look for two numbers whose product is +6 and sum is −5. These are −2 and −3, so:

寻找两个数，乘积为 +6，和为 −5。这两个数是 −

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

更多咨询请联系16621398022（同微信）