Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratics are the gateway to advanced algebra and a guaranteed topic in IGCSE Mathematics. This guide covers every method you need — factorisation, completing the square, the quadratic formula and graphical solutions — with common pitfalls flagged throughout.

二次方程是通往高级代数的大门，也是 IGCSE 数学中的必考内容。本指南涵盖你所需的一切解法——因式分解、配方法、二次公式和图像法——并全程标注常见误区。

1. The Standard Form | 标准形式

A quadratic equation is any equation that can be written in the form:

二次方程是任何可以写成以下形式的方程：

ax² + bx + c = 0

where a, b and c are constants, and a ≠ 0. The highest power of x is 2, which is why it is called a ‘quadratic’. If a = 0, the equation becomes linear and all the methods below simplify accordingly.

其中 a、b、c 为常数，且 a ≠ 0。x 的最高次数是 2，因此称为”二次方程”。若 a = 0，方程退化为一次方程，以下所有方法也会相应简化。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the expression factorises neatly. The idea is to rewrite ax² + bx + c as a product of two brackets, then use the zero-product property.

当表达式能整齐地分解时，因式分解是最快的方法。思路是将 ax² + bx + c 写成两个括号的乘积，然后利用零乘积性质。

Step-by-step example: Solve x² + 5x + 6 = 0.

逐步示例：解 x² + 5x + 6 = 0。

Find two numbers that multiply to 6 and add to 5: those are 2 and 3.

寻找两个乘积为 6、和为 5 的数，即 2 和 3。

(x + 2)(x + 3) = 0

Now apply the zero-product property: if the product of two factors is zero, at least one factor must be zero.

现在应用零乘积性质：若两个因式的乘积为零，则至少有一个因式为零。

x + 2 = 0 或 x + 3 = 0

x = −2 或 x = −3

Always check your answer by substituting back into the original equation.

务必把答案代回原方程检验。

3. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic in the form a(x + p)² + q, which makes it easy to solve and also reveals the vertex of the parabola.

配方法将二次方程改写为 a(x + p)² + q 的形式，既便于求解，也能揭示抛物线的顶点。

Worked example: Solve x² − 4x − 5 = 0.

例题：解 x² − 4x − 5 = 0。

Step 1: Move the constant to the right: x² − 4x = 5.

步骤一：将常数项移到右边：x² − 4x = 5。Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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