IGCSE Maths Topic T-1-1066: Quadratic Equations | IGCSE 数学专题 T-1-1066：二次方程

📚 IGCSE Maths Topic T-1-1066: Quadratic Equations | IGCSE 数学专题 T-1-1066：二次方程

Quadratic equations appear throughout the IGCSE syllabus, from algebraic manipulation and coordinate geometry to real-life modelling. A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a ≠ 0. This article covers factorising, completing the square, the quadratic formula, graphs, inequalities and common exam traps.

二次方程贯穿 IGCSE 大纲，从代数运算、坐标几何到现实建模均有涉及。二次方程是任何能写成标准形式 ax² + bx + c = 0 的方程，其中 a ≠ 0。本文涵盖因式分解、配方法、二次公式、图像、不等式以及常见考试陷阱。

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

In IGCSE Mathematics, a quadratic equation is an equation of degree 2. Its standard form is ax² + bx + c = 0, where a, b and c are constants and a cannot be zero. If a = 0, the equation becomes linear. Quadratic equations may have two real solutions, one repeated real solution, or no real solutions, depending on the discriminant, which we will discuss later.

在 IGCSE 数学中，二次方程是次数为 2 的方程。其标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a 不能为 0。若 a = 0，方程就变成一次方程。二次方程可能有两个实根、一个重根或没有实根，这取决于后面要讲的判别式。

You should be able to recognise quadratic equations even when they are not written neatly. For example, x² – 3x = 4 can be rearranged to x² – 3x – 4 = 0. Expanding products such as (x + 2)(x – 5) = 0 also gives a quadratic.

即使方程没有整齐写出，你也要能识别二次方程。例如 x² – 3x = 4 可以整理为 x² – 3x – 4 = 0。展开乘积如 (x + 2)(x – 5) = 0 也会得到二次方程。

2. Solving by Factorising | 因式分解法求解

Factorising is usually the fastest method when the quadratic has integer roots. First rewrite the equation in the form ax² + bx + c = 0. Then factorise the left-hand side into two linear brackets. Finally, set each bracket equal to zero and solve for x.

当二次方程有整数根时，因式分解通常是最快的方法。先把方程写成 ax² + bx + c = 0。然后把左边分解为两个一次因式。最后令每个括号等于零并解出 x。

Example: Solve x² – 5x + 6 = 0. Factorise: (x – 2)(x – 3) = 0. Thus x – 2 = 0 or x – 3 = 0, so x = 2 or x = 3.

例如：解 x² – 5x + 6 = 0。因式分解得 (x – 2)(x – 3) = 0。因此 x – 2 = 0 或 x – 3 = 0，所以 x = 2 或 x = 3。

3. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. It is useful for finding turning points, deriving the quadratic formula, and solving equations that do not factorise neatly. For a monic quadratic x² + bx + c, add and subtract (b/2)².

配方法把 ax² + bx + c 写成 a(x + p)² + q 的形式。它适用于求顶点、推导二次公式以及求解不易因式分解的方程。对于首项系数为 1 的二次式 x² + bx + c，加减 (b/2)²。

x² + bx + c = (x + b/2)² – (b/2)² + c

Example: Solve x² + 6x + 2 = 0 by completing the square.

例如：用配方法解 x² + 6x + 2 = 0。

x² + 6x = -2 → x² + 6x + 9 = 7 → (x + 3)² = 7 → x = -3 ± √7

We add 9 to both sides because (6/2)² = 9. The final solutions are x = -3 + √7 and x = -3 – √7.

我们在两边同时加 9，因为 (6/2)² = 9。最终解为 x = -3 + √7 和 x = -3 – √7。

4. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0. It states that

二次公式可以求解任何二次方程 ax² + bx + c = 0。公式为

x = [-b ± √(b² – 4ac)] / (2a)

This is given in the IGCSE formula sheet for many boards, but you must know how to substitute correctly.

许多考试局的 IGCSE 公式表会提供该公式，但你必须会正确代入。

Example: Solve 2x² – 3x –

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

更多咨询请联系16621398022（同微信）