IGCSE Mathematics: Solving Quadratic Equations | IGCSE 数学：解一元二次方程

📚 IGCSE Mathematics: Solving Quadratic Equations | IGCSE 数学：解一元二次方程

Quadratic equations are one of the most important algebra topics in IGCSE Mathematics, appearing in both Core and Extended papers. This article covers the standard form, the main methods of solution, and the link between algebraic roots and graphs.

一元二次方程是 IGCSE 数学中最重要的代数主题之一，在核心课程和扩展课程试卷中都会出现。本文将介绍二次方程的标准形式、主要求解方法，以及代数根与图像之间的联系。

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

A quadratic equation in one unknown is an equation that can be expressed as ax² + bx + c = 0, where a, b and c are constants and a is not zero. The term ‘quadratic’ comes from the idea of a square, because the highest power of x is 2.

一元二次方程是含有一个未知数、且可以表示为 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a 不等于零。“二次”的名称来源于平方，因为 x 的最高次数是 2。

At IGCSE level, you are expected to solve quadratic equations by factorising, completing the square and using the quadratic formula. You must also understand how the solutions connect to the x-intercepts of the graph of y = ax² + bx + c.

在 IGCSE 阶段，你需要会通过因式分解、配方法和求根公式解一元二次方程，还必须理解方程的解与函数 y = ax² + bx + c 图像上的 x 轴截距之间的联系。

2. Standard Form and Key Terms | 标准形式与关键术语

ax² + bx + c = 0, a ≠ 0

In the standard form, a is the coefficient of x², b is the coefficient of x, and c is the constant term. For example, in 2x² − 3x + 4 = 0, the coefficients are a = 2, b = −3 and c = 4.

在标准形式中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。例如，在 2x² − 3x + 4 = 0 中，系数为 a = 2，b = −3，c = 4。

• ax² is the quadratic term，即二次项。
• bx is the linear term，即一次项。
• c is the constant term，即常数项。

The value of a cannot be zero, because if a = 0 the equation becomes a linear equation, not a quadratic one.

a 的值不能为零，因为如果 a = 0，方程就变成了一次方程，而不是二次方程。

3. Solving by Factorisation | 因式分解法求解

Factorising is usually the fastest method when the quadratic has simple integer roots. First set the equation equal to zero, then write the quadratic expression as a product of two linear factors.

当二次方程有较简单的整数根时，因式分解

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