Solving Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程求解

📚 Solving Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程求解

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to real-world modelling. A quadratic equation is any equation that can be rearranged into the form ax² + bx + c = 0, where a ≠ 0. This article explains the main methods of solution, common errors and exam strategies.

二次方程在 IGCSE 数学课程中贯穿代数、图像和实际问题建模等模块。任何能整理成 ax² + bx + c = 0（其中 a ≠ 0）的方程都是二次方程。本文将讲解主要解法、常见错误和考试策略。

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

A quadratic equation is a polynomial equation of degree two. In IGCSE, it usually has one variable, often x, and the highest power of the variable is 2. The standard form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a were zero, the x² term would disappear and the equation would become linear rather than quadratic. For example, x² − 5x + 6 = 0 and 2x² + 3x − 2 = 0 are quadratic equations.

二次方程是次数为 2 的多项式方程。在 IGCSE 中，它通常只有一个变量，一般是 x，而且变量的最高次数是 2。标准形式为 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。如果 a = 0，x² 项就会消失，方程就变成一次方程而不是二次方程。例如 x² − 5x + 6 = 0 和 2x² + 3x − 2 = 0 都是二次方程。

The word ‘quadratic’ comes from the Latin ‘quadratus’, meaning square, because the variable is squared. The solutions of a quadratic equation are values of x that make the equation true. They are also called roots or zeros.

‘quadratic’ 一词来自拉丁文 ‘quadratus’，意思是平方，因为变量被平方了。二次方程的解就是使方程成立的 x 值，它们也被称为根或零点。

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

Before solving, always rearrange the equation into standard form ax² + bx + c = 0. Collect all terms on one side, simplify like terms, and write the x² term first, then the x term, then the constant. This makes factorising and identifying a, b and c much easier.

解题前，一定要先把方程整理成标准形式 ax² + bx + c = 0。把所有项移到一边，合并同类项，并按照先 x² 项、再 x 项、最后常数项的顺序书写。这样在因式分解以及确定 a、b、c 时会容易得多。

The coefficient a is the number multiplying x², b is the number multiplying x, and c is the constant term. When a = 1, the quadratic is called monic. The roots are the solutions of the equation, and they correspond to the x-intercepts of the graph y = ax² + bx + c.

系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。当 a = 1 时，这个二次式称为首一二次式。根是方程的解，它们对应图像 y = ax² + bx + c 与 x 轴的交点。

3. Solving by Factorising | 因式分解法

Factorising works when the quadratic expression can be written as the product of two linear factors. For example, x² − 5x + 6 = (x − 2)(x − 3). Since the product is zero, at least one factor must be zero, so x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

当二次式可以写成两个一次因式的乘积时，就可以使用因式分解法。例如 x² − 5x + 6 = (x − 2)(x − 3)。因为乘积为零，所以至少有一个因式为零，于是 x − 2 = 0 或 x − 3 = 0，解得 x = 2 或 x = 3。

For non-monic cases such as 2x² + 5x − 3, find two numbers that multiply to 2 × (−3) = −6 and add to 5: these are 6 and −1. Split the middle term: 2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (2x − 1)(x + 3). Then solve 2x − 1 = 0 or x + 3 = 0 to get x = ½ or x = −3.

对于非首一的情况，例如 2x² + 5x − 3，要找到两个数，使它们的乘积等于 2 × (−3) = −6，并且和等于 5：这两个数是 6 和 −1。将中间项拆分：2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (2x − 1)(x + 3)。然后解 2x − 1 = 0 或 x + 3 = 0，得到 x = ½ 或 x = −3。

The standard steps are: rearrange to ax² + bx + c = 0, factorise the quadratic expression, set each factor equal to zero, and solve each linear equation. Always check that the product of the factors expands back to the original expression.

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