Mastering Quadratic Equations for IGCSE | IGCSE 二次方程完全掌握

📚 Mastering Quadratic Equations for IGCSE | IGCSE 二次方程完全掌握

A quadratic equation is one of the most important algebraic skills in the IGCSE Mathematics syllabus. It appears in paper 1 and paper 2, often linked to graphs, inequalities, and real-life modelling problems. Mastering the methods below will give you a strong foundation for higher-level algebra.

二次方程是 IGCSE 数学大纲中最重要的代数技能之一。它常出现在试卷一和试卷二中，并且经常与图像、不等式以及实际建模问题结合考查。掌握下面的方法将为你打下扎实的高等代数基础。

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

A quadratic equation in one variable is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the unknown x is 2, which means the equation has a degree of 2.

一元二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 是常数，并且 a ≠ 0。未知数 x 的最高次数是 2，也就是说该方程的次数为 2。

For example, 2x² – 5x + 3 = 0 is a quadratic equation, but 3x – 7 = 0 is linear because the highest power of x is 1.

例如，2x² – 5x + 3 = 0 是二次方程，但 3x – 7 = 0 是一次方程，因为 x 的最高次数为 1。

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

Before solving, you should always rearrange the equation into standard form ax² + bx + c = 0. The coefficients a, b and c can be positive, negative, fractions or even irrational numbers.

在求解之前，你应当总是把方程整理成标准形式 ax² + bx + c = 0。系数 a、b 和 c 可以是正数、负数、分数，甚至无理数。

• a is the coefficient of x².
• b is the coefficient of x.
• c is the constant term.

• a 是 x² 的系数。
• b 是 x 的系数。
• c 是常数项。

If a = 1, the quadratic is called monic. For example, x² + 6x + 8 = 0 is monic, while 3x² + 6x + 8 = 0 is not.

如果 a = 1，这个二次方程称为首一二次方程。例如，x² + 6x + 8 = 0 是首一的，而 3x² + 6x + 8 = 0 不是。

3. Solving by Factorising | 因式分解法

Factorising is usually the fastest method when the quadratic has simple integer roots. The idea is to write ax² + bx + c as the product of two linear brackets, then use the zero product property: if p × q = 0, then p = 0 or q = 0.

当二次方程有简单的整数根时，因式分解通常是最快的方法。其思路是将 ax² + bx + c 写成两个一次括号的乘积，然后利用零乘积性质：如果 p × q = 0，则 p = 0 或 q = 0。

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

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

For non-monic quadratics such as 2x² + 7x + 3 = 0, you can use the grouping method: find two numbers that multiply to ac = 6 and add to b = 7, namely 6 and 1. Then rewrite 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1) = 0, so x = -3 or x = -1/2.

对于非首一二次方程，例如 2x² + 7x + 3 = 0，你可以使用分组法：找到两个数，使它们的积等于 ac = 6，和等于 b = 7，即 6 和 1。然后将原式改写为 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1) = 0，所以 x = -3 或 x = -1/2。

4. Solving by Completing the Square | 配方法

Completing the square is a powerful method because it works for every quadratic equation and it also helps you find the vertex of a parabola. The key identity is x² + bx = (x + b/2)² – (b/2)².

配方法是一种强有力的方法，因为它适用于所有二次方程，并且还能帮助你找到抛物线的顶点。关键恒等式是 x² + bx = (x + b/2)² – (b/2)²。

Example: Solve x² + 6x – 7 = 0. Rewrite as x² + 6x = 7. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16. Then (x + 3)² = 16, so x + 3 = ±4, giving x = 1 or x = -7.

例题：解 x² + 6x – 7 = 0。改写为 x² + 6x = 7。两边同时加上 (6/2)² = 9：x² + 6x + 9 = 16。于是 (x + 3)² = 16，所以 x + 3 = ±4，得到 x = 1 或 x = -7。

5. The Quadratic Formula | 二次公式法

When factorising is difficult, the quadratic formula gives the exact roots of ax² + bx + c = 0. The formula is:

当因式分解困难时，二次公式可以给出 ax² + bx + c = 0 的精确根。公式为：

x = (-b ± √(b² – 4ac)) / 2a

Example: Solve 2x² + 3x – 2 = 0. Here a = 2, b

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