Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from factorising simple trinomials to using the quadratic formula and reading key features from a parabola. This article builds the topic step by step and gives you the tools to solve examination questions confidently.

二次方程在 IGCSE 数学课程中随处可见，从简单的因式分解三项式到使用二次求根公式，再到从抛物线图像中读取关键信息。本文会循序渐进地构建这一主题，帮助你自信地解答考试题目。

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

A quadratic equation is a polynomial equation in which the highest power of the variable is two. It can always be written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a were zero, the equation would become linear rather than quadratic.

二次方程是未知量最高次幂为 2 的多项式方程。它总能写成一般形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。如果 a 为 0，方程就变成一次方程而不是二次方程。

IGCSE questions may give a quadratic in a disguised form, such as x² = 5x – 6. Your first job is to rearrange every term to one side so the other side equals zero. Only then can you identify the standard form and choose a suitable solving method.

IGCSE 考题有时会给出隐藏形式的二次方程，例如 x² = 5x – 6。你的第一步是把所有项移到一边，使另一边等于 0。只有整理成标准形式后，你才能识别系数并选择适当的求解方法。

2. Standard Form and Coefficients | 标准形式与系数

The standard form ax² + bx + c = 0 tells us the names of the coefficients: a is the coefficient of x², b is the coefficient of x, and c is the constant term. Identifying a, b and c correctly is essential before using the quadratic formula or discriminant.

标准形式 ax² + bx + c = 0 明确了系数的名称：a 是 x² 的系数，b 是 x 的系数，c 是常数项。在使用二次公式或判别式之前，正确识别 a、b、c 至关重要。

For example, in 3x² – 7x + 2 = 0, we have a = 3, b = -7 and c = 2. Notice that the sign stays with the coefficient: b is negative because the term is -7x. Similarly, in 4x – x² = 6, rearrange to -x² + 4x – 6 = 0, so a = -1, b = 4 and c = -6.

例如，在 3x² – 7x + 2 = 0 中，a = 3，b = -7，c = 2。注意符号要跟着系数：因为这一项是 -7x，所以 b 是负数。类似地，在 4x – x² = 6 中，要整理为 -x² + 4x – 6 = 0，因此 a = -1，b = 4，c = -6。

• Write the equation in the form ax² + bx + c = 0 first. 先写为 ax² + bx + c = 0。
• Keep negative signs attached to the correct coefficient. 负号要紧跟对应的系数。
• Never ignore a negative value of a. 不要忽略 a 的负号。

3. Solving by Factorising | 因式分解法求解

Factorising is usually the fastest method when the quadratic has integer roots. After rearranging to standard form, look for two numbers that multiply to give ac and add to give b, or use the simpler case where a = 1 and the numbers multiply to c and add to b.

当二次方程有整数根时，因式分解通常是最快的方法。整理成标准形式后，寻找两个数，使它们相乘等于 ac 且相加等于 b；当 a = 1 时，更简单，只需找到相乘等于 c 且相加等于 b 的两个数。

Solve x² – 5x + 6 = 0. We need two numbers whose product is +6 and sum is -5: -2 and -3. Therefore (x – 2)(x – 3) = 0. Setting each factor to zero gives x = 2 or x = 3.

求解 x² – 5x + 6 = 0。我们需要两个数，乘积为 +6，和为 -5：这两个数是 -2 和 -3。因此 (x – 2)(x – 3) = 0。令每个因式为零，得到 x = 2 或 x = 3。

For quadratics where a is not 1, such as 2x² + 5x

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