Quadratic Equations: Solving, Graphing and Applications | 二次方程：求解、图像与应用

📚 Quadratic Equations: Solving, Graphing and Applications | 二次方程：求解、图像与应用

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from factorising simple trinomials to interpreting curved graphs in real-life problems. This revision guide brings together the methods, graphs and applications you need to answer both core and extended questions confidently.

二次方程贯穿 IGCSE 数学考纲，从简单的三项式因式分解到结合实际问题的曲线图像都会考查。本复习指南汇总了考试所需的解法、图像与应用，帮助你更自信地应对核心卷和扩展卷题目。

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

A quadratic equation is a polynomial equation of degree 2. In one variable, it can always be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is essential because if a were zero the equation would become linear.

二次方程是次数为 2 的多项式方程。在一元情形下，它总可以写成 ax² + bx + c = 0 的形式，其中 a、b、c 为常数且 a ≠ 0。a ≠ 0 这个条件很关键，因为如果 a 为零，方程就变成了一次方程。

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.

y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上并有一个最低点；若 a < 0，抛物线开口向下并有一个最高点。

ax² + bx + c = 0, a ≠ 0

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

Before solving, you must rearrange a quadratic into standard form. For example, x² = 5x − 6 becomes x² − 5x + 6 = 0. The coefficient a is called the leading coefficient, b is the linear coefficient and c is the constant term.

求解前必须先把二次方程整理成标准形式。例如 x² = 5x − 6 要写成 x² − 5x + 6 = 0。系数 a 称为二次项系数，b 是一次项系数，c 是常数项。

The solutions of a quadratic equation are called its roots. On a graph, the roots are the x-coordinates where the parabola crosses the x-axis. These points are also called x-intercepts or zeros.

二次方程的解称为根。在图像上，根就是抛物线与 x 轴交点的横坐标。这些点也叫 x 轴截距或零点。

Always check that the right-hand side is zero before factorising or using the formula. This is the most common first step in IGCSE quadratic questions.

在因式分解或使用求根公式之前，一定要检查等式右边是否为零。这是 IGCSE 二次方程题中最常见的第一步。

3. Solving by Factorisation | 因式分解法

Factorisation works when the quadratic can be written as a product of two linear brackets. For x² − 5x + 6 = 0, we find two numbers that multiply to give c = 6 and add to give b = −5. The numbers are −2 and −3

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